Common Core Math: High School Algebra: Polynomials & Rational Expressions

A monomial is the product of a number (called a coefficient) and variables raised to nonnegative integer powers (e.g. 4x2). A polynomial is a mathematical expression that can be written as the sum of one or more monomials (e.g. 4x2 + 5x3). In high school, students learn to compose polynomials via addition, subtraction and multiplication and to decompose polynomials via factoring and division. A rational expression is a ratio of two polynomials. Students extend their knowledge of polynomials to manipulate and perform arithmetic with rational expressions. In the course of their studies of polynomial arithmetic, students learn and apply two key theorems: the binomial theorem and the remainder theorem.

Common Core Standards

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CCSS.Math.Content.HSA-APR.A.1
high school polynomials common core standard C.5

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rationals and polynomials common core standards

Factor Polynomials

Decompose and analyze polynomials in terms of their factors.

Factor polynomials (CCSS.Math.Content.HSA-APR.C.4):

factor x^2 - 4
x^2 + 8x + 12 factored
factor x^3 + 8
x^4 - 9 in factored form

Find zeros of a polynomial (CCSS.Math.Content.HSA-APR.B.3):

zeros of (x + 2)(x - 5)
roots of x(2x - 5)(x + 4)
real roots of 8x^5 + 32x^4 + 5x + 20
factor x^2 + 2x - 3
y = x^2 - 4 when y = 0
0 = x^3 + 2x^2 + x + 2

Recognize and apply polynomial identities (CCSS.Math.Content.HSA-APR.C.4):

factor a^2 - b^2
factor (x^2 - y^2)^2 + (2xy)^2
factor a^3 + b^3

Apply the remainder theorem (CCSS.Math.Content.HSA-APR.B.2):

remainder theorem
f(2) if f(x) = x^2 + 5x + 6
x^3 + 3x^2 + 2x + 8 at x = -3

Add, Subtract & Multiply Polynomials

Combine polynomials via addition, subtraction or multiplication.

Identify polynomials (CCSS.Math.Content.HSA-APR.A.1):

is x^2 + x + 1 a polynomial?
is y = sqrt(x) a polynomial?

Add and subtract polynomials (CCSS.Math.Content.HSA-APR.A.1):

(2x + 3x^2) + (4x + 5 - x^2)
(4x^2 + 3x + 6) - (2x^2 - 5x - 2)
a^2 + 4b + 6a + 4a^2 + 5ab + 5b + 3a^2
(x^2 + 4x + 5) + (8x + 8) + 7

Multiply polynomials (CCSS.Math.Content.HSA-APR.A.1):

expand (x + 7) * (x - 8)
(x^2 + 4x + 4) (x^2 - 1) expanded form
multiply out (x + 1) (x - 1) (x + 2)
(x + 1) * (2x + 5) - (3x^2 + 2x - 1)
expanded form of (x^2 + x)^4
(2x + 5) (3x^2 + 4x + 10)

Apply the binomial theorem (CCSS.Math.Content.HSA-APR.C.5):

4th term in (4x + y)^4
coefficient of x^2 in (2x + 1)^5
expand (x + y)^3
(2x - 1)^4 expanded form
compute binomial coefficient
Pascal's triangle

Rational Expressions

Rewrite and compute with ratios of polynomials.

Perform polynomial division (CCSS.Math.Content.HSA-APR.D.6):

(x^2 + 5x + 6)/(x + 3)
x^2 - 9x + 18 divided by x - 6
(x^4 - 8x^3 + 10x^2 + 2x + 4)/(x - 2)
quotient and remainder of (x^2 + 5x + 6)/(x + 4)

Perform arithmetic with rational expressions (CCSS.Math.Content.HSA-APR.D.7):

(a + b)/a + b/(4a)
x/(x^2 - 1) - 2x/(x^2 + x -2)
((2a - 12)/(3b + 4))^2 * ((a^2 - 2a - 3)/(b^2 + 4b + 4))
[(x + 5)/(x^2 + 7x + 10)]/[(x^2 + 8x + 12)/(x + 2)]
((x^2 + x - 1)/(2x - 1) - 1)/(4 - 8/(b + 1))
k/(k + 1) + 1/((k + 1) (k + 2))
1/(1 + 1/(1 + 1/(1 + 1/x)))

相关示例

  • Common Core Math: Creating Equations
  • Common Core Math: Expressions & Equations
  • Common Core Math: Reasoning with Equations & Inequalities
  • Common Core Math: Seeing Structure in Expressions