Factoring a Polynomial. Example 1: Factoring a Polynomial Completely factor x 3 + 2x 2 – 11x – 12 Use the graph or table to find at least one real root.

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Presentation transcript:

Factoring a Polynomial

Example 1: Factoring a Polynomial Completely factor x 3 + 2x 2 – 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).

Example 1: Factoring a Polynomial Completely factor x 3 + 2x 2 – 11x – 12 x 3 + 2x 2 – 11x – 12 x + 4 x2x2 x3x3 4x24x2 -2x 2 -2x -8x -3x Now we can rewrite the cubic: Since the graph of the cubic had more than one real root, this may be able to be factored more. This quadratic can be factored using old techniques: (x + 1)(x – 3) Thus, the completely factored form is:

Let’s try another example.

Example 2: Factoring a Polynomial Completely factor x 4 – x 3 + 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).

Example 2: Factoring a Polynomial x 4 – x 3 + 0x 2 + 4x – 16 x + 2 x3x3 x4x4 2x32x3 -3x 3 -3x 2 -6x 2 6x26x2 6x6x 12x -8x -8 Completely factor x 4 – x 3 + 4x – Now we can rewrite the degree 4 equation: Make sure to include all powers of x Since the graph of the degree 4 equation had more than one real root, this may be able to be factored more. Let’s check the graph of this cubic to see if it has a real root.

Example 2: Factoring a Polynomial Completely factor x 4 – x 3 + 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.

Example 2: Factoring a Polynomial x 3 – 3x 2 + 6x – 8 x – 2 x2x2 x3x3 -2x 2 -x2-x2 -x-x 2x2x 4x4x 4 -8 Now we can rewrite the current factored form as: Since the graph of the cubic had only one real root, this may NOT be able to be factored more. This quadratic can NOT be factored using old techniques (No x-intercepts). Thus, the completely factored form is: Current Factored form: Completely factor x 4 – x 3 + 4x – 16