The Remainder and Factor Theorems 6.5 p. 352. When you divide a Polynomial f(x) by a divisor d(x), you get a quotient polynomial q(x) with a remainder.

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Presentation transcript:

The Remainder and Factor Theorems 6.5 p. 352

When you divide a Polynomial f(x) by a divisor d(x), you get a quotient polynomial q(x) with a remainder r(x) written: f(x) = q(x) + r(x) d(x) d(x)

The degree of the remainder must be less than the degree of the divisor!

Polynomial Long Division: You write the division problem in the same format you would use for numbers. If a term is missing in standard form …fill it in with a 0 coefficient. Example: 2x 4 + 3x 3 + 5x – 1 = x 2 – 2x + 2

2x 4 = 2x 2 x 2 2x 2

+4x 2 -4x 3 2x 4 -( ) - 4x 2 7x 3 +5x 7x 3 = 7x x 2 +7x 7x x 2 +14x-( ) 10x 2 - 9x x x +20-( ) 11x - 21 remainder

The answer is written: 2x 2 + 7x x – 21 x 2 – 2x + 2 Quotient + Remainder over divisor

Now you try one! y 4 + 2y 2 – y + 5 = y 2 – y + 1 Answer: y 2 + y y 2 – y + 1

Remainder Theorem: If a polynomial f(x) is divisible by (x – k), then the remainder is r = f(k). Now you will use synthetic division (like synthetic substitution) f(x)= 3x 3 – 2x 2 + 2x – 5 Divide by x - 2

f(x)= 3x 3 – 2x 2 + 2x – 5 Divide by x - 2 Long division results in ? x 2 + 4x x – 2 Synthetic Division: f(2) = Which gives you: 3x 2 + 4x x-2

Synthetic Division Practice 1 Divide x 3 + 2x 2 – 6x -9 by (a) x-2 (b) x+3 (a) x Which is x 2 + 4x x-2

Synthetic Division Practice cont. (b) x x 2 – x - 3

Factor Theorem: A polynomial f(x) has factor x-k iff f(k)=0 note that k is a ZERO of the function because f(k)=0

Factoring a polynomial Factor f(x) = 2x x x + 9 Given f(-3)=0 Since f(-3)=0 x-(-3) or x+3 is a factor So use synthetic division to find the others!!

Factoring a polynomial cont (x + 3)(2x 2 + 5x + 3) So…. 2x x x + 9 factors to: Now keep factoring (bustin ‘da ‘b’) gives you: (x+3)(2x+3)(x+1)

Your turn! Factor f(x)= 3x x 2 + 2x -8 given f(-4)=0 (x + 1)(3x – 2)(x + 4)

Finding the zeros of a polynomial function f(x) = x 3 – 2x 2 – 9x +18. One zero of f(x) is x=2 Find the others! Use synthetic div. to reduce the degree of the polynomial function and factor completely. (x-2)(x 2 -9) = (x-2)(x+3)(x-3) Therefore, the zeros are x=2,3,-3!!!

Your turn! f(x) = x 3 + 6x 2 + 3x -10 X=-5 is one zero, find the others! The zeros are x=2,-1,-5 Because the factors are (x-2)(x+1)(x+5)

Assignment