October 11 th copyright2009merrydavidson Happy August Birthday to: Collin Stipe.

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Remainder and Factor Theorems

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Presentation transcript:

October 11 th copyright2009merrydavidson Happy August Birthday to: Collin Stipe

If you divide 12 by 3 the remainder is zero, therefore 3 is a “factor” of 12. Divide (x 4 – 1) by (x + 1) What is the remainder? Conclusion: (x + 1) is a FACTOR of (x 4 – 1). zero

The Remainder Theorem The value of a function at a specific “x” is the same as the remainder when divided by that value. FORf(x) = 3x 3 + 8x 2 + 5x -7 FIND f(-2) = Now use Synthetic Division for x = -2 3(-2) 3 +8(-2) 2 +5(-2)-7=9 What was the remainder?9

Practice Find the remainder if: f(x) = x 3 – 4x 2 + 2x –5 is divided by (x – 3). Do you need to perform division? Find f(3). -8 No you can use direct substitution.

The Factor Theorem If the binomial (x – r) is a factor of f(x) then f(r ) = 0 What does this mean? If (x – 7) is a factor, Then f(7) = 0

Practice Is (x – 5) a factor of P(x) = x 3 – 4x 2 – 7x + 10? yes

Practice Is (x + 2) a factor of f(x) = 5x 3 + 8x 2 – x + 6? no

Show that (x+3)is a factor of f(x) = x 3 + 4x 2 + x – 6 Then find the remaining factors. Step 1: divide f(x) by (x+3) to show it has a remainder of zero Step 2: factor the remaining quadratic. Step 3: Write the polynomial in factored form.

Show that (x-2) and (x+3)are factors of f(x) = 2x 4 + 7x 3 – 4x 2 – 27x -18 Then find the remaining factors. Step 1: divide f(x) by (x-2) Step 2: divide the depressed polynomial by (x+3). Step 3: factor the remaining quadratic. Step 4: Write the polynomial in factored form.

Homework WS 3-6