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Factoring a Polynomial

Factoring a Polynomial. Example 1: Factoring a Polynomial. Completely factor x 3 + 2 x 2 – 11 x – 12. Use the graph or table to find at least one real root. . x = -4 is a real root because it is an x-intercept. Since x = -4 is a root, (x + 4) is a factor of the original cubic equation.

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Factoring a Polynomial

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  1. Factoring a Polynomial

  2. Example 1: Factoring a Polynomial Completely factor x3+ 2x2– 11x – 12 Use the graph or table to find at least one real root. x = -4 is a real root because it is an x-intercept. Since x = -4 is a root, (x + 4) is a factor of the original cubic equation. Now use polynomial division to “factor out” the (x + 4).

  3. Example 1: Factoring a Polynomial Completely factor x3+ 2x2– 11x – 12 Now we can rewrite the cubic: x2 -2x -3 x x3 -2x2 -3x This quadratic can be factored using old techniques: (x + 1)(x – 3) + 4 4x2 -8x -12 Since the graph of the cubic had more than one real root, this may be able to be factored more. x3 + 2x2– 11x – 12 Thus, the completely factored form is:

  4. Let’s try another example.

  5. Example 2: Factoring a Polynomial Completely factor x4 – x3 + 4x – 16 Use the graph or table to find at least one real root. x = -2 is a real root because it is an x-intercept. Since x = -2 is a root, (x + 2) is a factor of the original degree 4 equation. Now use polynomial division to “factor out” the (x + 2).

  6. Example 2: Factoring a Polynomial Completely factor x4 – x3 + 4x – 16 Now we can rewrite the degree 4 equation: x3 -3x2 6x -8 x x4 -3x3 6x2 -8x Let’s check the graph of this cubic to see if it has a real root. + 2 2x3 -6x2 12x -16 Since the graph of the degree 4 equation had more than one real root, this may be able to be factored more. x4 – x3 + 0x2 + 4x – 16 Make sure to include all powers of x

  7. Example 2: Factoring a Polynomial Completely factor x4 – x3 + 4x – 16 Current Factored form: Use the graph or table of the cubic in the factored form to find at least one real root. x = 2 is a real root because it is an x-intercept. Since x = 2 is a root, (x – 2) is a factor of the cubic in the factored form. Now use polynomial division to “factor out” the (x – 2) of the cubic in the factored form.

  8. Example 2: Factoring a Polynomial Completely factor x4 – x3 + 4x – 16 Current Factored form: Now we can rewrite the current factored form as: x2 -x 4 x x3 -x2 4x This quadratic can NOT be factored using old techniques (No x-intercepts). – 2 -2x2 2x -8 Since the graph of the cubic had only one real root, this may NOT be able to be factored more. x3 – 3x2 + 6x – 8 Thus, the completely factored form is:

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