How do you write $P (x) = x^{3} + 6 x^{2} - 15 x - 100$ $P(x) = x^3 + 6x^2 − 15x − 100$ in factored form?

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How do you write $P (x) = x^{3} + 6 x^{2} - 15 x - 100$ $P(x) = x^3 + 6x^2 − 15x − 100$ in factored form?

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Answer 1 · 2015-09-13T13:57:05

$x^{3} + 6 x^{2} - 15 x - 100 = (x - 4) (x + 5) (x + 5)$ $x^3+6x^2-15x-100 = (x-4)(x+5)(x+5)$

Explanation:

By the rational root theorem, any rational roots of $P (x) = 0$ $P(x) = 0$ must be expressible in the form $\frac{p}{q}$ $p/q$ , where $p, q in ZZ$ , $q > 0$ , $"hcf"(p, q) = 1$ , $p$ a divisor of the constant term $-100$ and $q$ a divisor of the coefficient $1$ of the leading term.

So the only possible rational roots of $P(x) = 0$ are:

$+-1$ , $+-2$ , $+-4$ , $+-5$ , $+-10$ , $+-25$ , $+-50$ , $+-100$

Let's try some:

$P(1) = 1+6-15-100 = -108$
$P(-1) = -1+6+15-100 = -80$
$P(2) = 8+24-30-100 = -102$
$P(-2) = -8+24+30-100 = -54$
$P(4) = 64+96-60-100 = 0$

So $(x-4)$ is a factor of $P(x)$ .

Use synthetic division to divide $P(x)$ by $(x-4)$ ...

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So $x^3+6x^2-15x-100 = (x-4)(x^2+10x+25)$

Now we can recognise $x^2+10x+25$ as a perfect square trinomial. It is of the form $a^2+2ab+b^2 = (a+b)^2$ , with $a = x$ and $b = 5$ .

So $x^2+10x+25 = (x+5)^2$

Putting this together:

$x^3+6x^2-15x-100 = (x-4)(x+5)(x+5)$

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How do you write $P (x) = x^{3} + 6 x^{2} - 15 x - 100$ $P(x) = x^3 + 6x^2 − 15x − 100$ in factored form?

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Explanation:

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How do you write P(x) = x^3 + 6x^2 − 15x − 100P(x)=x3+6x2−15x−100P(x) = x^3 + 6x^2 − 15x − 100 in factored form?

1 Answer

Explanation:

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How do you write $P (x) = x^{3} + 6 x^{2} - 15 x - 100$ $P(x) = x^3 + 6x^2 − 15x − 100$ in factored form?