Polynomial and Synthetic Division
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1 Chapter Polynomial Functions. f y. Common function: y Transformation: Vertical stretch each y-value is multiplied by, then a vertical shift nine units upward f Horizontal shift of three units to the left, vertical shrink each y-value is multiplied by, reflection in the -ais and vertical shift of ten units upward of y 7 y 7 Section. Polynomial and Synthetic Division You should know the following basic techniques and principles of polynomial division. The Division Algorithm (Long Division of Polynomials) Synthetic Division f k is equal to the remainder of f divided by k. (The Remainder Theorem) f k if and only if k is a factor of f. Vocabulary Check. f is the dividend; d is the divisor; g is the quotient; r is the remainder. improper; proper. synthetic division. factor. remainder. y and ) y Thus, and y y.. y and ) y Thus, and y y.. y and y and (b) ) Thus, and y y.
2 Section. Polynomial and Synthetic Division. y and y and (b) ) Thus, and y y.. ). 7 ) 7 7. ) ) ) 7 ) 7 7 7
3 Chapter Polynomial Functions. ) ). ). ). ). ) ). ) 7. 7
4 Section. Polynomial and Synthetic Division
5 Chapter Polynomial Functions.. 7. f, k. f, k f f 7 f f. f, k f f. f, k f 7 f 7. f, k. f, k f f f f. f, k f f. f, k f f
6 Section. Polynomial and Synthetic Division. f. g f g (b) (d) f f f (b) g (d) g g 7. h. (b) h 7 h h f...7 (b) f. f f (d) (d) h f.. 7 Zeros:,, Zeros:,,
7 Chapter Polynomial Functions. Zeros:,, 7 7. Zeros:,,. Zeros: ±,. Zeros:,,. Zeros:, ±. Zeros:,, 7 7. f ; Factors:, Both are factors of f since the remainders are zero. (b) The remaining factor of f is. f,, 7
8 Section. Polynomial and Synthetic Division. f ; Factors:, (b) The remaining factor is. f,,. f ; Factors:, (b) Both are factors of f since the remainders are zero. The remaining factors are and. f,,,. f 7 ; Factors:, (b) The remaining factors are and.. f ; Factors:, Both are factors since the remainders are zero. f 7 f,,, (b) 7 This shows that so The remaining factor is 7.,, f 7, f 7.
9 Chapter Polynomial Functions. f 7 ; Factors: (b) This shows that so The remaining factor is., f. f,, f,. f ; Factors:, Both are factors since the remainders are zero. (b) This shows that so The remaining factor is. f f.,, f,. f ; Factors:, f ±, (b) The remaining factor is.. f. g The zeros of f are and (b) An eact zero is. f ±.. The zeros of g are,., (b) is an eact zero. f..
10 Section. Polynomial and Synthetic Division 7 7. h t t t 7t. f s s s s The zeros of h are t, t.7, t.. The zeros of f are s, s.7, s. (b) An eact zero is t. (b) s is an eact zero. h t t t t By the Quadratic Formula, the zeros of t t are ±. Thus, h t t t t f s s s s s s s t t t.. Thus,,. 7. 7, 7.,, , ± 7. and (b) CONTINUED
11 Chapter Polynomial Functions 7. CONTINUED M.t.t 7.t Year, t Military Personnel M The model is a good fit to the actual data. (d) M thousand No, this model should not be used to predict the number of military personnel in the future. It predicts an increase in military personnel until and then it decreases and will approach negative infinity quickly. 7. and (b) 7. False. If is a factor of f, then 7 is a zero of f. (b) R.t.t.t For the year, the model predicts a monthly rate of about $.. 7. True. f 77. True. The degree of the numerator is greater than the degree of the denominator. 7. f k q r k, r, q any quadratic a b c where a >. One eample: f (b) k, r, q any quadratic a b c where a <. One eample: f
12 Section. Polynomial and Synthetic Division 7. n n n ) n n 7 n 7. n n n ) n n n n n n 7 n n n n 7 n 7 n n n n n n n n n n 7 n 7 n n n n n n n n n. A divisor divides evenly into a dividend if the remainder is zero.. You can check polynomial division by multiplying the quotient by the divisor. This should yield the original dividend if the multiplication was performed correctly.. To divide evenly, equal. c c c must equal zero. Thus, c must. c c To divide evenly, c must equal zero. Thus, c must equal.. f The remainder when k is zero since is a factor of f.. In this case it is easier to evaluate f directly because f is in factored form. To evaluate using synthetic division you would have to epand each factor and then multiply it all out. 7.. ± ± or or. b ± b ac a ± ± ±
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Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 1 of 3 Topic 5: Quadratic Functions (Part 1) Definition: A quadratic function is a function which can be written as f x ax bx
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How might we evaluate this? Suppose that, by some good luck, we knew that. x 2 5. x 2 dx 5
8.4 1 8.4 Partial Fractions Consider the following integral. 13 2x (1) x 2 x 2 dx How might we evaluate this? Suppose that, by some good luck, we knew that 13 2x (2) x 2 x 2 = 3 x 2 5 x + 1 We could then