5.5 Apply the Remainder and Factor Theorems

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Remainder and Factor Theorems

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Presentation transcript:

5.5 Apply the Remainder and Factor Theorems Algebra II

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: 2x4 + 3x3 + 5x – 1 = x2 – 2x + 2

2x2 2x4 = 2x2 x2

-( ) 2x2 +7x +10 2x4 -4x3 +4x2 7x3 - 4x2 +5x -( ) 7x3 - 14x2 +14x -( ) 2x4 -4x3 +4x2 7x3 - 4x2 +5x -( ) 7x3 - 14x2 +14x 10x2 - 9x -1 7x3 = 7x x2 -( ) 10x2 - 20x +20 11x - 21 remainder

The answer is written: 2x2 + 7x + 10 + 11x – 21 x2 – 2x + 2 Quotient + Remainder over divisor

Now you try one! y4 + 2y2 – y + 5 = y2 – y + 1 Answer: y2 + y + 2 + 3 y2 – 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)= 3x3 – 2x2 + 2x – 5 Divide by x - 2

f(x)= 3x3 – 2x2 + 2x – 5 Divide by x - 2 Long division results in ?...... 3x2 + 4x + 10 + 15 x – 2 Synthetic Division: f(2) = 3 -2 2 -5 2 6 8 20 3 4 10 15 Which gives you: + 15 x-2 3x2 + 4x + 10

Synthetic Division Practice 1 Divide x3 + 2x2 – 6x -9 by (a) x-2 (b) x+3 (a) x-2 1 2 -6 -9 8 2 4 1 4 2 -5 Which is x2 + 4x + 2 + -5 x-2

Synthetic Division Practice cont. (b) x+3 1 2 -6 -9 -3 3 -3 9 1 -1 -3 x2 – x - 3

5-5B 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) = 2x3 + 11x2 + 18x + 9 Given x + 3 Since f(-3)=0 x-(-3) or x+3 is a factor So use synthetic division to find the others!!

Factoring a polynomial cont. 2 11 18 9 -3 -15 -9 -6 2 5 3 So…. 2x3 + 11x2 + 18x + 9 factors to: (x + 3)(2x2 + 5x + 3) Now keep factoring gives you: (x+3)(2x+3)(x+1)

Ex. 3 ) Your turn! Factor f(x)= 3x3 + 13x2 + 2x -8 given x+4

Finding the zeros of a polynomial function f(x) = x3 – 2x2 – 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)(x2-9) = (x-2)(x+3)(x-3) Therefore, the zeros are x=2,3,-3!!!

Your turn! f(x) = x3 + 6x2 + 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