1.4 - Factoring Polynomials - The Remainder Theorem

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1.4 - Factoring Polynomials - The Remainder
Theorem

• MCB4U - Santowski

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(A) Review

• to evaluate a polynomial for a given value of x,
  we simply substitute that given value of x into
  the polynomial.
• ex. Evaluate 4x3 - 6x² x - 3 for x 2
• we then have a special notations that we can
  write
• for the polynomial P(x) 4x3 - 6x² x - 3
• for substituting into the polynomial P(2)
  4(2)3 6(2)² (2) - 3

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(B) The Remainder Theorem

• Divide 3x3 4x2 - 2x - 5 by x 1
• Evaluate P(-1). What do you notice?
• if rewritten as 3x3 4x2 - 2x - 5 (x 1)(3x5
  - 7x 5) - 10, notice P(-1) -10 (Why?)
•  
• Divide 6p2 - 17p - 7 by 3p 1
• Evaluate P(-1/3). What do you notice? Rewrite the
  equation in factored form
•  
• Divide 8p2 - 11p 5 by 2p - 5
• Evaluate P(5/2). What do you notice? What must be
  true about (2p-5)?
•  
• Divide x2 - 5x 4 by x - 4
• Evaluate P(-4). What do you notice? What must be
  true about (x - 4)?

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(B) The Remainder Theorem

• the remainder theorem states "when a polynomial,
  P(x), is divided by (ax - b), and the remainder
  contains no term in x, then the remainder is
  equal to P(b/a)

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(C) Examples

• Find k so that when x2 8x k is divided by x -
  2, the remainder is 3
• Find the value of k so that when x3 5x2 6x
  11 is divided by x k, the remainder is 3
• When P(x) ax3 x2 - x b is divided by x - 1,
  the remainder is 6. When P(x) is divided by x
  2, the remainder is 9. What are the values of a
  and b?

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(D) Internet Links

• Remainder Theorem and Factor Theorem from WTAMU
• The Remainder Theorem from The Math Page
• Remainder Theorem Lesson From Purple Math

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(E) Homework

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• Nelson text, page 50, Q2,3,4,9 (verify using RT),
  10,11,14