How do you determine an equation of a polynomial function with zeros at x=2,-2,1 and y-intercept of 24?

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How do you determine an equation of a polynomial function with zeros at x=2,-2,1 and y-intercept of 24?

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Answer 1 · 2016-02-18T00:33:02

$y = x^{4} + 5 x^{3} - 10 x^{2} - 20 x + 24$ $y=x^4+5x^3-10x^2-20x+24$

Explanation:

If ${2, - 2, 1}$ ${2,-2,1}$ are all zeros of the polynomial
then
the polynomial must contain the factors:
$XXX (x - 2) (x + 2) (x - 1)$ $color(white)("XXX")(x-2)(x+2)(x-1)$

So
$XXX y = (x - 2) (x + 2) (x - 1) \times a$ $color(white)("XXX")y=(x-2)(x+2)(x-1)xxa$ for some additional factor $a$ $a$

$¯ y = (x - 2) (x + 2) (x - 1) = x^{3} - x^{2} - 4 x + 4$ $bary = (x-2)(x+2)(x-1)=x^3-x^2-4x+4$
which is equal to $4$ $4$ when $x = 0$ $x=0$
So if $y = ¯ y \times a$ $y=baryxxa$ is to be equal to $24$ $24$ when $x = 0$ $x=0$
then $a$ $a$ must have the value $6$ $6$ when $x = 0$ $x=0$

The most obvious (but not only) value for $a$ $a$ is
$XXX (x + 6)$ $color(white)("XXX")(x+6)$
in which case
$XXX y = (x - 2) (x + 2) (x - 1) (x + 6)$ $color(white)("XXX")y=(x-2)(x+2)(x-1)(x+6)$
$XXXX = x^{4} + 5 x^{3} - 10 x^{2} - 20 x + 24$ $color(white)("XXXX")=x^4+5x^3-10x^2-20x+24$

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How do you determine an equation of a polynomial function with zeros at x=2,-2,1 and y-intercept of 24?

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