Synthetic division and the factor theorem

Question

Use the factor theorem to determine if the binomials given are factors of f(x). Use the binomials that are factors to write a factored form of f(x).

f(x)=3x^4-4x^3-61x^2+22x+40;(3x+2),(x-1)

Answer 1

Need f(a) = 0 for (x-a) to be a factor

f(-2/3) = 16/27 + 32/27 - 732/27 - 396/27 - 1080/27 = 0/27 = 0 √

f(1) = 3 - 4 - 61 + 22 + 40 = 0 √

3 -4 -61 22 40 |1

3 -1 -62 -40

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3 -1 -62 -40 |-2/3

-2 2 40

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3 -3 -60 --> 3x² -3x -60 --> x² - x - 20 easy factor --> (x+4)(x-5)

f(x) = (3x+2)(x-1)(x+4)(x-5)