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#### MAT-HS.A-APR

Domains are larger groups of related standards. So the Domain Score is a calculation of all the related standards. So click on the standard name below each Domain to access the learning targets and proficiency scales for each Domain's related standards.

# Arithmetic with Polynomials & Rational Expressions

• Perform arithmetic operations on polynomials
• Understand the relationship between zeros and factors of polynomials
• Use polynomial identities to solve problems
• Rewrite rational functions

## Domain Description

An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances.

Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05p can be interpreted as the addition of a 5% tax to a price p. Rewriting p + 0.05p as 1.05p shows that adding a tax is the same as multiplying the price by a constant factor.

Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example, p + 0.05p is the sum of the simpler expressions p and 0.05p. Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure.

A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave.

## Standards in this Domain

• MAT-HS.A-APR.01 - Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
• MAT-HS.A-APR.02 - Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).
• MAT-HS.A-APR.03 - Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
• MAT-HS.A-APR.04 - Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.
• MAT-HS.A-APR.05 - Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.
• MAT-HS.A-APR.06 - Rewrite simple rational expressions in different forms; write a(x)b(x) in the form q(x) + r(x)b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
• MAT-HS.A-APR.07 - Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

#### MAT-HS.A-APR.01

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Perform arithmetic operations on polynomials MAT-HS.A-APR.01 Add, subtract, and multiply polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.

## Student Learning Targets:

### Knowledge Targets

• I can identify polynomials.

### Reasoning Targets

• I can recognize how closure applies under these operations.
• I can

### Skills (Performance) Targets

• I can add, subtract, and multiply polynomials and recognize how closure applies under these operations.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. Shannon, a cabinetmaker, started out with a block of wood, and then she hollowed out the center of the block.  The dimensions of the block and the cutout is shown below. Write the volume of the original block, and the volume of the hole. (ignore units for this problem) Write the polynomial for the volume of the wood remaining. 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: add, subtract, and multiply polynomials. recognize that when two polynomials are added, subtracted, or multiplied, a polynomial is the result. The student exhibits no major errors or omissions. Simplify the following: (2x2+3x+5)+(5x3-9x-8) (x4+3x3-x2)-(4x3+8) (x+2)(5x2-7x+10) 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student can: recognize and recall specific terminology, such as: polynomial trinomial binomial term combine like terms. However, the student exhibits major errors or omissions regarding the more complex ideas and processes. Simplify the following: 2x2+7x-9+3x2+5x-10 (2x+3)(4x-8) 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill.

## Resources

 Web Vocab Polynomial

#### MAT-HS.A-APR.02

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Perform arithmetic operations on polynomials MAT-HS.A-APR.02 Apply the Remainder Theorem.

## Student Learning Targets:

### Knowledge Targets

• I can use the fact that a is a zero of a polynomial.

• I can
• I can

### Skills (Performance) Targets

• I can apply the Remainder Theorem to a polynomial.

• I can
• I can

## Alg II Solve Polynomial Equations Proficiency Scale

Score   Description Sample Activity

4.0

In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations.
 Describe and correct the error in using synthetic division to divide x3-5x+3 by x-2.
3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success.

3.0

(proficient)

The student can:
• use and apply the remainder theorem/synthetic division to find the zeros of a polynomial function, real and imaginary.
• use and apply long division/synthetic division to find the zeros of a polynomial function.
The student exhibits no major errors or omissions.

Find the zeros:

f(x)= x3-10x2+19x+30

f(x)=2x5+3x4-30x3-57x2-2x+24

f(x)=x3-4x2+25x-100
2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content).

2.0

(progressing)

There are no major errors or omissions regarding the simpler details and processes as the student can:
• recognize and recalls specific terminology, such as:
• factors
• zeros
• imaginary/complex numbers
• synthetic division or long division
• fundamental theorem of Algebra
• possible zeros (p/q)
• use synthetic division to divide polynomials.
• can perform operations with complex numbers.

However, the student exhibits major errors or omissions regarding the more complex ideas and processes.

Synthetically divide to find the remaining zeros.

(6x3-19x2+16x-4)(x-2)

(2x4+7x3-4x2-27x-18)(x+3)
1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content).

1.0

(beginning)

With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content).
0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).
0.0 Even with help, the student demonstrates no understanding or skill.

## Resources

 Web Vocab

#### MAT-HS.A-APR.03

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Perform arithmetic operations on polynomials MAT-HS.A-APR.03 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. a. Substandard b. Substandard

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can generate graphs of a variety of functions (i.e., linear, quadratic, polynomial, absolute value, and exponential), using technology when appropriate.
• I can solve for (isolate) a specific variable.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.).
• I can check the solution.
• I can set a polynomial equal to zero and solve by factoring, quadratic formula, graphing, linear combinations, and substitution.
• I can solve for (isolate) a specific variable.
• I can write a linear system.
• I can perform transformations that produce equivalent equations (e.g., adding the same amount to both sides of the equation, etc.)
• I can check the solution.
• I can solve a quadratic.
• I can multiply and factor polynomials.
• I can solve a linear inequality.
• I can graph a linear inequality.
• I can solve a system of linear inequalities.
• I can graph a system of linear inequalities.
• I can Check the solution of the system.
• I can solve a linear system (substitution, linear combination, graphically).
• I can write a linear system.
• I can identify the number of solutions of a linear system.
• I can identify the most efficient method to solve a system.
• I can check the solution of the system.
• I can find the zeros of a polynomial function when the polynomial is factored.

### Product Targets

• I can use the zeros (x-intercepts) of a polynomial function to sketch a graph of the function.

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-APR.04

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Use polynomial identities to solve problems MAT-HS.A-APR.04 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)² = (x² – y²)² + (2xy)² can be used to generate Pythagorean triples.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can use polynomial identities such as those for x2 – y2, x3 – y3, x3 + y3, (x+y)3 and (x+y)2 to describe numerical relationships.
• I can prove polynomial identities.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-APR.05

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Use polynomial identities to solve problems. MAT-HS.A-APR.05 Know and apply the Binomial Theorem gives the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can use Pascal’s Triangle to determine the coefficients of the binomial expansion (x+y)n.
• I can use the Binomial Theorem to find the nth term in the expansion of a binomial to a positive integer power.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-APR.06

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Rewrite rational expressions. MAT-HS.A-APR.06 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

## Student Learning Targets:

• I can
• I can

• I can
• I can

### Skills (Performance) Targets

• I can rewrite rational expressions, a(x)/b(x), in the form q(x) + r(x)/b(x) using inspection or long division (use a computer algebra system for complicated examples).

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-APR.07

 MAT-HS Targeted Standards(A) Concept: Algebra(SSE) Domain: Arithmetic with Polynomials and Rational ExpressionsCluster: Use polynomial identities to solve problems MAT-HS.A-APR.07 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

## Student Learning Targets:

### Knowledge Targets

• I can recognize that rational expressions are closed under addition, subtraction, multiplication, and division by a nonzero expression.

• I can
• I can

### Skills (Performance) Targets

• I can add, subtract, multiply, and divide rational expressions.

• I can
• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 In addition to Score 3.0, the student demonstrates in-depth inferences and applications regarding more complex material that go beyond end of instruction expectations. - 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 “The Standard.” The student demonstrates no major errors or omissions regarding any of the information and processes that were end of instruction expectations. - 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 The student demonstrates no major errors or omissions regarding the simpler details and processes but exhibits major errors or omissions regarding the more complex ideas and processes (Score 3.0 content). - 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) and some of the more complex ideas and processes (Score 3.0 content). - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content). 0.0 Even with help, the student demonstrates no understanding or skill. -

## Resources

 Web Vocab

#### MAT-HS.A-CED

Domains are larger groups of related standards. So the Domain Score is a calculation of all the related standards. So click on the standard name below each Domain to access the learning targets and proficiency scales for each Domain's related standards.

# Creating Equations

• Create equations that describe numbers or relationships

## Domain Description

An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form.

The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system.

An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.

Some equations have no solutions in a given number system, but have a solution in a larger system. For example, the solution of x + 1 = 0 is an integer, not a whole number; the solution of 2x + 1 = 0 is a rational number, not an integer; the solutions of  – 2 = 0 are real numbers, not rational numbers; and the solutions of  + 2 = 0 are complex numbers, not real numbers.

The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b1+b2)/2)h, can be solved for h using the same deductive process. Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.

Connections to Functions and Modeling. Expressions can define functions, and equivalent expressions define the same function. Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling.

## Standards in this Domain

• MAT-HS.A-CED.01 - Create equations and inequalities in one variable and use them to solve problems.  Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
• MAT-HS.A-CED.02 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
• MAT-HS.A-CED.03 - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
• MAT-HS.A-CED.04 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

#### MAT-HS.A-CED.01.a

 MAT-HS Targeted Standards(A) Concept: Algebra(CED) Domain: Creating ExpressionsCluster: Create equations that describe numbers or relationships MAT-HS.A-CED.01.a Create equations in one variable and use them to solve problems (solving is tied to the REI standards). Include equations arising from linear functions.

## Student Learning Targets:

• I can

• I can

### Skills (Performance) Targets

• I can create a linear equation in one variable given a situation or set of data.

• I can

## Proficiency Scale

 Score Description Sample Activity 4.0 (advanced) In addition to achieving level 3.0 content, the student makes in-depth inferences and applications that go beyond what was taught It takes Bill 6 minutes to wash a car. It takes Jason 8 minutes to wash a car. Write and solve an equation to represent the amount of time it would take Bill and Jason to wash the car together. 3.5 In addition to Score 3.0 performance, the student demonstrates in-depth inferences and applications regarding the more complex content with partial success. 3.0 (proficient) The student can: Can create equations and inequalities in one variable and use them to solve problems (linear only) The student exhibits no major errors or omissions. An online music club sells compact discs for $13.95 each plus$1.95 shipping and handling per order.  Maria’s total bill was \$85.65.  Create an equation to model this situation.  Solve your equation to determine how many compact discs were sold. 2.5 The student demonstrates no major errors or omissions regarding the simpler details and processes (Score 2.0 content) and partial knowledge of the more complex ideas and processes (Score 3.0 content). 2.0 (progressing) There are no major errors or omissions regarding the simpler details and processes as the student: recognizes and recalls specific terminology, such as: linear correctly identifies whether the relationship is linear or not write general equations with one variable pick out relevant information in a contextual problem However, the student exhibits major errors or omissions regarding the more complex ideas and processes. The sum of a number t and 8 is 7. 1.5 The student demonstrates partial knowledge of the simpler details and processes (Score 2.0 content) but exhibits major errors or omissions regarding the more complex ideas and procedures (Score 3.0 content). 1.0 (beginning) With help, a partial understanding of some of the simpler details and processes and some of the more complex ideas and processes. - 0.5 With help, the student demonstrates a partial understanding of some of the simpler details and processes (Score 2.0 content) but not the more complex ideas and processes (Score 3.0 content).

## Resources

 Web Vocab linear equations quadratic equations rational equations exponential equations inequalities

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