4. Polynomial and Rational Functions

Use Synthetic Division to Divide Polynomials

Dividing Polynomials

❖ Synthetic Division - A Shortcut for Long Division! ❖

In Exercises 53–54, write a polynomial that represents the length of each rectangle. 

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Using the Remainder Theorem to Evaluate a Polynomial Video

In Exercises 17–32, divide using synthetic division. (x^7+x^5−10x^3+12)/(x+2)

In Exercises 17–32, divide using synthetic division. (x^5+4x^4−3x^2+2x+3)÷(x−3)

Find k so that 4x+3 is a factor of 20x^3+23x^2-10x+k.

Perform each division. See Examples 9 and 10. (p^2+2p+20)/(p+6)

Solve the equation 12x^3+16x^2−5x−3=0 given that -3/2 is a root.

In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x^3−11x^2+7x−5;f(4)

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x^3+13x^2−11x−15)/(3x^2−x−3)

In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=x^4+5x^3+5x^2−5x−6;f(3)

In Exercises 17–32, divide using synthetic division. (3x^2+7x−20)÷(x+5)

Perform each division. See Examples 9 and 10. (x^2+11x+16)/(x+8)

Perform each division. See Examples 7 and 8. (-4m^2n^2-21mn^3+18mn^2)/(-14m^2n^3)

The Remainder Theorem - Example 1

In Exercises 17–32, divide using synthetic division. (2x^5−3x^4+x^3−x^2+2x−1)/(x+2)

Use synthetic division to show that 5 is a solution of x^4−4x^3−9x^2+16x+20=0. Then solve the polynomial equation.

Using the remainder theorem to evaluate a polynomial

In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=6x^4+10x^3+5x^2+x+1;f(− 2/3)

Comprehensive Graph of a Polynomial Function

Perform each division. See Examples 7 and 8. (8wxy^2+3wx^2y+12w^2xy)/(4wx^2y)

Perform each division. See Examples 9 and 10. (3x^3-2x+5)/(x-3)

The Remainder Theorem - Example 2

In Exercises 17–32, divide using synthetic division. (6x^5−2x^3+4x^2−3x+1)÷(x−2)

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (x^4−81)/(x−3)

Perform each division. See Examples 9 and 10. (4x^3-3x^2+1)/(x-2)

In Exercises 17–32, divide using synthetic division. (x^2−5x−5x^3+x^4)÷(5+x)

In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x^4−5x^3−x^2+3x+2;f(− 1/2)

How to do Synthetic Division

In Exercises 17–32, divide using synthetic division. (x^5+x^3−2)/(x−1)

Perform each division. See Examples 9 and 10. (q^2+4q-32)/(q-4)

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (3x^2−2x+5)/(x−3)

Given f(x) = 2x^3 - 7x^2 + 9x - 3, use the Remainder Theorem to find f(- 13).

Divide a polynomial by a binomial using long division

Relate factors and zeros using the Factor Theorem

In Exercises 17–32, divide using synthetic division. (4x^3−3x^2+3x−1)÷(x−1)

In Exercises 27–29, divide using long division. (4x^4 +6x^3 + 3x - 1) ÷ (2x^2 + 1)

❖ Long Division of Polynomials ❖

Perform each division. See Examples 9 and 10. (10x^3+11x^2-2x+3)/(5x+3)

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x^3+7x^2+12x−5)÷(3x−1)

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (12x^2+x−4)÷(3x−2)

Divide polynomials using long division

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (x^4+2x^3−4x^2−5x−6)/(x^2+x−2)

Perform each division. See Examples 9 and 10. (4x^3+9x^2-10x-6)/(4x+1)

When 2x^2−7x+9 is divided by a polynomial, the quotient is 2x-3 and the remainder is 3. Find the polynomial.

Synthetic Division

Long Division of Polynomials - A slightly harder example

In Exercises 27–29, divide using long division. (4x^3 - 3x^2 - 2x + 1) ÷ (x + 1)

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (2x^5+9x^3+3x^2)/(3x^2+1)

In Exercises 17–32, divide using synthetic division. (x^4−256)/(x−4)

In Exercises 17–32, divide using synthetic division. (2x^2+x−10)÷(x−2)

Apply the Remainder Theorem to simple division problems

Perform each division. See Examples 9 and 10. (3t^2+17t+10)/(3t+2)

Perform each division. See Examples 9 and 10. (x^4+5x^2+5x+27)/(x^2+3)

Solve the equation 2x3−3x2−11x+6=0 given that -2 is a zero of f(x)=2x^3−3x^2−11x+6.

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (x^2+8x+15)÷(x+5)

Perform each division. See Examples 7 and 8. (-4x^7-14x^6+10x^4-14x^2)/(-2x^2)

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (x^3+5x^2+7x+2)÷(x+2)

Solving Higher Degree Polynomials by Synthetic Division and the Rational Roots Test

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (4x^4−4x^2+6x)/(x−4)

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (2x^3+7x^2+9x−20)/(x+3)

Divide 737 by 21 without using a calculator. Write the answer as quotient + remainder/divisor

Find the roots of a polynomial using a graphing calculator

Perform each division. See Examples 7 and 8. (15m^3+25m^2+30m)/(5m^3)

In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=3x^3−7x^2−2x+5;f(−3)

Find solutions to polynomial equations where the polynomial's remainder is zero

Use synthetic division to divide f(x)=x^3−4x^2+x+6 by x+1. Use the result to find all zeros of f.

Divide polynomials with a missing term

Perform each division. See Examples 7 and 8. (9y2+12y-5)/(3y)

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College Algebra

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1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

5. Exponential and Logarithmic Functions

6. Systems of Equations and Inequalities

7. Matrices and Determinants

8. Conic Sections

9. Sequences, Induction, and Probability

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Use Long Division to Divide Polynomials

Evaluate a Polynomial Using the Remainder Theorem

Use the Factor Theorem to Solve a Polynomial Equation

Identify Polynomial Functions

Recognize Characteristics of Graphs of Polynomial Functions

Determine End Behavior

Use Factoring to Find Zeros of Polynomial Functions

Identify Zeros and Their Multiplicities

Use the Intermediate Value Theorem

Understand the Relationship Between Degree and Turning Points