# High School: Algebra » Arithmetic with Polynomials & Rational Expressions

Print this page## Standards in this domain:

#### Perform arithmetic operations on polynomials.

CCSS.Math.Content.HSA.APR.A.1

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.

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.

#### Understand the relationship between zeros and factors of polynomials.

CCSS.Math.Content.HSA.APR.B.2

Know and apply the Remainder Theorem: For a polynomial

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*).CCSS.Math.Content.HSA.APR.B.3

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.

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.

#### Use polynomial identities to solve problems.

CCSS.Math.Content.HSA.APR.C.4

Prove polynomial identities and use them to describe numerical relationships.

Prove polynomial identities and use them to describe numerical relationships.

*For example, the polynomial identity (x*^{2}+ y^{2})^{2}= (x^{2}- y^{2})^{2}+ (2xy)^{2}can be used to generate Pythagorean triples.CCSS.Math.Content.HSA.APR.C.5

(+) Know and apply the Binomial Theorem for the expansion of (

(+) Know and apply the Binomial Theorem for 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.^{1}#### Rewrite rational expressions.

CCSS.Math.Content.HSA.APR.D.6

Rewrite simple rational expressions in different forms; write

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.CCSS.Math.Content.HSA.APR.D.7

(+) 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.

(+) 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.

^{1}
The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.