How do you factor given that $f (4) = 0$ $f(4)=0$ and $f (x) = x^{3} - 14 x^{2} + 47 x - 18$ $f(x)=x^3-14x^2+47x-18$ ?

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How do you factor given that $f (4) = 0$ $f(4)=0$ and $f (x) = x^{3} - 14 x^{2} + 47 x - 18$ $f(x)=x^3-14x^2+47x-18$ ?

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Answer 1 · 2016-08-02T18:51:56

$f (x) = (x - 9) (x - \frac{5}{2} - \frac{\sqrt{17}}{2}) (x - \frac{5}{2} + \frac{\sqrt{17}}{2})$ $f(x) = (x-9)(x-5/2-sqrt(17)/2)(x-5/2+sqrt(17)/2)$

Explanation:

The difference of squares identity can be written:

$a^{2} - b^{2} = (a - b) (a + b)$ $a^2-b^2 = (a-b)(a+b)$

We use this later with $a = (x - \frac{5}{2})$ $a=(x-5/2)$ and $b = \frac{\sqrt{17}}{2}$ $b=sqrt(17)/2$ .

By the rational root theorem, any rational zeros of $f (x)$ $f(x)$ must be expressible in the form $\frac{p}{q}$ $p/q$ for integers $p, q$ $p, q$ with $p$ $p$ a divisor of the constant term $- 18$ $-18$ and $q$ $q$ a divisor of the coefficient $1$ $1$ of the leading term.

So the only possible rational zeros are:

$\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$ $+-1, +-2, +-3, +-6, +-9, +-18$

In particular $f (4) \neq 0$ $f(4) != 0$ since $4$ $4$ is not a factor of $18$ $18$ .

Trying each of these in turn, we (eventually) find:

$f (9) = 729 - 1134 + 423 - 18 = 0$ $f(9) = 729-1134+423-18 = 0$

So $(x - 9)$ $(x-9)$ is a factor of $f (x)$ $f(x)$ :

$x^{3} - 14 x^{2} + 47 x - 18$ $x^3-14x^2+47x-18$

$= (x - 9) (x^{2} - 5 x + 2)$ $= (x-9)(x^2-5x+2)$

$= (x - 9) ({(x - \frac{5}{2})}^{2} - \frac{25}{4} + 2)$ $= (x-9)((x-5/2)^2-25/4+2)$

$= (x - 9) ⎛ ⎝ {(x - \frac{5}{2})}^{2} - {(\frac{\sqrt{17}}{2})}^{2} ⎞ ⎠$ $= (x-9)((x-5/2)^2-(sqrt(17)/2)^2)$

$= (x - 9) (x - \frac{5}{2} - \frac{\sqrt{17}}{2}) (x - \frac{5}{2} + \frac{\sqrt{17}}{2})$ $= (x-9)(x-5/2-sqrt(17)/2)(x-5/2+sqrt(17)/2)$

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How do you factor given that $f (4) = 0$ $f(4)=0$ and $f (x) = x^{3} - 14 x^{2} + 47 x - 18$ $f(x)=x^3-14x^2+47x-18$ ?

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Explanation:

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How do you factor given that f(4)=0f(4)=0f(4)=0 and f(x)=x^3-14x^2+47x-18f(x)=x3−14x2+47x−18f(x)=x^3-14x^2+47x-18?

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Explanation:

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How do you factor given that $f (4) = 0$ $f(4)=0$ and $f (x) = x^{3} - 14 x^{2} + 47 x - 18$ $f(x)=x^3-14x^2+47x-18$ ?