Lesson 2.4, page 301 Dividing Polynomials

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Lesson 2.4, page 301Dividing Polynomials

• Objective To divide polynomials using long and
  synthetic division, and to use the remainder and
  factor theorems.

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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.

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Dividing a Poly by a Binomial
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See Example 1, page 302.

• Check Point 1(x2 14x 45) ? (x 9)

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Check Point 2(7 11x - 3x2 2x3) ? (x - 3)
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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.

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Check Point 3(2x4 3x3 7x - 10) ? (x2 2x)
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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.

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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.

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Check Point 4, page 307

• Use Synthetic Division x3 7x 6 by x 2.
• Caution What is missing?

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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

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See Example 5, pg. 308

• Check Point 5
• Given f(x) 3x3 4x2 5x 3, use the
  remainder theorem to find f(- 4).

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Dividing a Poly by a Binomial

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

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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.

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See Example 6, pg. 309.

• Check Point 6 Solve the equation
• 15x3 14x2 3x 2 0, given that -1
• is a zero of f(x) 15x3 14x2 3x 2.

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Determine if -1 is a zero ofg(x) x4 - 6x3 x2
24x -20.