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2.5 Apply the Remainder and Factor Theorems p. 120 How do you divide polynomials? What is the remainder theorem? What is the difference between synthetic substitution and synthetic division? What is the factor theorem?

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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)

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

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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

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

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

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

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Now you try one! y 4 + 2y 2 – y + 5 = y 2 – y + 1 Answer: y 2 + y + 2 + 3 y 2 – y + 1

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2. (x 3 – x 2 + 4x – 10) (x + 2) SOLUTION Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x 2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

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Multiply divisor by x 3 /x = x 2. x 3 + 2x 2 –3x 2 + 4x Subtract. Bring down next term. Multiply divisor by – 3x 2 /x = –3x. – 3x 2 – 6x 10x – 1 Subtract. Bring down next term. Multiply divisor by 10x/x = 10. 10x + 20 – 30 remainder x 2 – 3x + 10 x + 2 x 3 – x 2 + 4x – 10 ) quotient

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x 3 – x 2 +4x – 10 x + 2 = (x 2 – 3x +10)+ – 30 x + 2 ANSWER OR…

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Use Synthetic Division (x 3 – x 2 + 4x – 10) (x + 2) Set x + 2 = 0. Solve for xx = −2 Use − 2 as the divisor for synthetic division which is the same as synthetic substitution. Synthetic division can be used to divide any polynomial by a divisor of the form “x −k”

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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

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F(x) = x 3 – x 2 + 4x – 10 (x + 2) – 2 1 −1 4 −10 – 2 6 – 20 1 – 3 10 – 30 ANSWER SOLUTION

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f(x)= 3x 3 – 2x 2 + 2x – 5 Divide by x - 2 Long division results in ?...... 3x 2 + 4x + 10 + 15 x – 2 Synthetic Division: f(2) = 3-22-5 2 3 6 4 8 10 20 15 Which gives you: 3x 2 + 4x + 10 + 15 x-2

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Synthetic Division Divide x 3 + 2x 2 – 6x -9 by (a) x-2 (b) x+3 (a) x-2 12-6-9 2 1 2 4 8 2 4 -5 Which is x 2 + 4x + 2 + -5 x-2

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Synthetic Division Practice cont. (b) x+3 12-6-9 -3 1 -3 3 -3 9 0 x 2 – x - 3

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

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Factoring a polynomial Factor f(x) = 2x 3 + 11x 2 + 18x + 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!!

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Factoring a polynomial cont. 211189 -3 2 -6 5 -15 3 -9 0 (x + 3)(2x 2 + 5x + 3) So…. 2x 3 + 11x 2 + 18x + 9 factors to: Now keep factoring-- gives you: (x+3)(2x+3)(x+1)

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Your Turn… Factor the polynomial completely given that x – 4 is a factor. f (x) = x 3 – 6x 2 + 5x + 12 SOLUTION Because x – 4 is a factor of f (x), you know that f (4) = 0. Use synthetic division to find the other factors. 4 1 – 6 5 12 4 – 8 –12 1 – 2 – 3 0

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Use the result to write f (x) as a product of two factors and then factor completely. f (x) = x 3 – 6x 2 + 5x + 12 Write original polynomial. = (x – 4)(x 2 – 2x – 3) Write as a product of two factors. = (x – 4)(x –3)(x + 1) Factor trinomial.

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Your turn! Factor f(x)= 3x 3 + 13x 2 + 2x -8 given f(-4)=0 (x + 1)(3x – 2)(x + 4)

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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!!!

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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)

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How do you divide polynomials? By long division What is the remainder theorem? If a polynomial f(x) is divisible by (x – k), then the remainder is r = f(k). What is the difference between synthetic substitution and synthetic division? It is the same thing What is the factor theorem? If there is no remainder, it is a factor.

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Assignment Page 124, 7, 9, 11-15 odd, 21-23 odd, 29-33 odd, 35- 37 all

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