Your track coach tells you to run a path around the campus. You are instructed to start at (1, 5) by the gym and run to (6, -2) which is the football field. Then halfway between the football field and the health field at (-2, 4), you are to stop for water. Finally, run back to the starting point at (1,5). What is the location of the water stop between the football field and the health field?

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Ответ:

belle200163

30.12.2021

Mathematics

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

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