Use the Factor Theorem and synthetic division to show x + 5 is a factor of f(x) = 2×3 + 7×2 − 14x + 5

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Use the Factor Theorem and synthetic division to show x + 5 is a factor of f(x) = 2×3 + 7×2 − 14x + 5

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Ngọc Hoa 4 years 2021-07-19T23:58:34+00:00 1 Answers 1 views 0

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  1. Latifah
    0
    2021-07-20T00:00:05+00:00
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    Answer:

    Factor theorem: f(-5) = 0.

    Synthetic division: f(x) = (x + 5)\, (2\, x^2 -3\, x + 1).

    Step-by-step explanation:

    Factor Theorem

    By the factor theorem, a monomial of the form (x - a) (where a is a constant) is a factor of polynomial f(x) if and only if f(a) = 0.

    In this question, the monomial is (x + 5), which is equivalently (x - (-5)). a = -5.

    \begin{aligned}& f(-5) \\ &= 2 \times (-5)^{3} + 7 \times (-5)^{2}- 14\times (-5) + 5\\ &= -250 + 175 + 70 + 5 \\ &= 0\end{aligned}.

    Hence, by the factor theorem,  (x + 5), which is equivalent to (x - (-5)), is a factor of f(x).

    Synthetic Division

    \begin{aligned}& f(x) \\ &= 2\, x^{3} + 7\, x^{2} - 14\, x + 5 \\ &= \underbrace{(x + 5) \, (2\, x^2)}_{2\, x^{3} + 10\, x^{2}} - 3\, x^{2} - 14\, x + 5 \\ &= \underbrace{(x + 5) \, (2\, x^2)}_{2\, x^{3} + 10\, x^{2}} + \underbrace{(x + 5)\, (-3\, x)}_{-3\, x^{2} - 15\, x} + (x + 5) \\ &= (x + 5)\, (2\, x^{2} - 3\, x + 1)\end{aligned}.

    The remainder is 0 when dividing f(x) by (x + 5). Hence, (x + 5)\! is a factor of f(x)\!.

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