Lesson 2.4, page 301Dividing Polynomials

Objective: To divide polynomials using long and

synthetic division, and to use the remainder and factor

theorems.

How do you divide a polynomial by another polynomial?

Perform long division, as you do with numbers! Remember, division is repeated subtraction, so each time you have a new term, you must SUBTRACT it from the previous term.

Work from left to right, starting with the highest degree term.

Just as with numbers, there may be a remainder left. The divisor may not go into the dividend evenly.

Dividing a Poly by a Binomial

24 33482

See Example 1, page 302.

Check Point 1:(x2 + 14x + 45) (x + 9)

Check Point 2(7 – 11x - 3x2 + 2x3) (x - 3)

Missing Terms?

Write the polynomial in standard form.

If any power is missing, use a zero to hold the place of that term.

Divide as before.

Check Point 3(2x4 + 3x3 – 7x - 10) (x2 – 2x)

Synthetic Division

a simpler process (than long division) for dividing a polynomial by a binomial; uses coefficients

and part of the divisor

See Example 4, page 306.

STEPS for Synthetic Division, pg. 306

1) Write polynomial in descending order of the degrees.2) List the coefficients. (If one power is missing, put a

zero to hold that place.)3) Write the constant c of the divisor x - c to the left. 4) Bring down the first coefficient.5) Multiply the first coefficient by c, write the product

under the 2nd coefficient and add.6) Multiply this sum by c, write it under the next

coefficient and add. Repeat until all coefficients have been used.

7) The numbers on the bottom row are the coefficients of the answer. The first power on the variable will be one less than the highest power in the original polynomial.

Check Point 4, page 307

Use Synthetic Division: x3 – 7x – 6 by x + 2.

Caution: What is missing?

The Remainder Theorem, pg. 307

If the polynomial f(x) is divided by x – c, then the remainder is the same

value as f(c).

Also:f(x) = (x – c) q(x) + r divisor (quotient) + remainder

See Example 5, pg. 308

Check Point 5:Given f(x) = 3x3 + 4x2 – 5x + 3, use the

remainder theorem to find f(- 4).

Dividing a Poly by a Binomial

If a binomial divides into a polynomial with no remainder, the binomial is a factor of the polynomial.

Factor Theorem, pg. 308

For the polynomial f(x), if f(c) = 0,

then x – c is a factor of f(x)

Remember . . . If something is a factor, then it divides the term evenly with 0 remainder.

See Example 6, pg. 309.

Check Point 6: Solve the equation15x3 + 14x2 – 3x – 2 = 0, given that -

1 is a zero of f(x) = 15x3 + 14x2 – 3x –

2.

Determine if -1 is a zero ofg(x) = x4 - 6x3 + x2 + 24x -

20.