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Factoring Polynomials of Higher Degree

Factoring Polynomials of Higher Degree. The Factor Theorem Part II. Factoring Polynomials of Higher Degree. From last class, synthetic division is a quick process to divide polynomials by binomials of the form x – a and bx – a. 2. 1 -4 -7 10. 2. -4. -22. 1. -2. -11. -12.

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Factoring Polynomials of Higher Degree

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  1. Factoring Polynomials of Higher Degree The Factor Theorem Part II

  2. Factoring Polynomials of Higher Degree • From last class, synthetic division is a quick process to divide polynomials by binomials of the form x – a and bx – a. 2 1 -4 -7 10 2 -4 -22 1 -2 -11 -12

  3. Factoring Polynomials of Higher Degree Example 1: Use synthetic division to find the quotient and remainder when 6x3 – 25x2 – 29x + 25 is divided by 2x – 1. Solution: 6 -25 -29 25 bx – a is written in the form in front of the “L”. 3 -11 -20 6 -22 -40 5

  4. Factoring Polynomials of Higher Degree The Factor Theorem (Part 2) If p(x) = anxn + an-1xn-1 + an-2xn-2 +…+ axx2 + azx + a0 is a polynomial function with integer coefficients, and if bx – a is a factor of p(x) where “b” and “a” are also integers, then “b” is a factor of the leading coefficient an and “a” is a factor of the constant term a0. “b” and “a” should have no factors in common.

  5. Factoring Polynomials of Higher Degree Example 2: Factor x3 + 5x2 + 2x – 8. Solution: If x3 + 5x2 + 2x – 8 has a factor of the form bx – a, then “a” is a factor of -8 and “b” is a factor of 1. A list of possible factors is x – 1, x – 2, x – 4, x – 8, x + 1, x + 2, x + 4, x + 8.  We will use synthetic division to try and find one factor. Start by testing x – 1.

  6. Factoring Polynomials of Higher Degree 1 1 5 2 -8 1 6 8 We note that x – 1 must be a factor since the remainder is 0. 1 6 8 0 Thus, we have: x3 + 5x2 + 2x – 8 = (x – 1)(x2 + 6x + 8) = (x – 1)(x + 2)(x +4) Factor the trinomial

  7. Factoring Polynomials of Higher Degree • Note that only 3 of the possible 8 solutions for the polynomial were factors. • It is unlikely that you will actually find a factor on your first try. • It will take a few attempts to find the first one, and then factor the remaining quadratic by inspection.

  8. Factoring Polynomials of Higher Degree Summary To factor a polynomial of higher degree: • Write the polynomial in decreasing degree inserting 0’s for any missing terms. • Make a list of all possible factors bx – a, where “a” is a factor of the constant term and “b” is a factor of the leading coefficent. “b” and “a” should have no factors in common. • Test the list of potential factors using synthetic division. • When you find a factor, factor the remaining quadratic.

  9. Homework • Do # 19, 25, 29, 31, 33, and 35 on page 130 from section 4.3 for Monday April 20th Have a Safe and Fun Easter Break 

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