SOLUTION: Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms. Third-degree, with zeros of −3, −1, and 4, and a y-intercept of −7.

Question 1197930: Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms.
Third-degree, with zeros of −3, −1, and 4, and a y-intercept of −7.
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

If k is a root of a polynomial, then x-k is a factor of that polynomial.

The roots -3, -1, 4 lead to factors x-(-3), x-(-1), and x-4
Simplifying the factors leads to (x+3), (x+1), (x-4)

So far we have
y = (x+3)(x+1)(x-4)

Stick an 'a' out front to be the leading coefficient
y = a(x+3)(x+1)(x-4)

The y-intercept is -7
Meaning x = 0 leads to y = -7
We'll use these values to find 'a'
y = a(x+3)(x+1)(x-4)
-7 = a(0+3)(0+1)(0-4)
-7 = a(3)(1)(-4)
-7 = -12a
a = -7/(-12)
a = 7/12

We go from
y = a(x+3)(x+1)(x-4)
to
y = (7/12)(x+3)(x+1)(x-4)

Then let's expand that out
y = (7/12)(x+3)(x+1)(x-4)
y = (7/12)(x+3)(x^2-4x+1x-4)
y = (7/12)(x+3)(x^2-3x-4)
y = (7/12)( x(x^2-3x-4)+3(x^2-3x-4) )
y = (7/12)( x^3-3x^2-4x+3x^2-9x-12 )
y = (7/12)( x^3-13x-12 )
y = (7/12)x^3+(7/12)(-13x)+(7/12)(-12 )
y = (7/12)x^3-(91/12)x-7

You can use a graphing tool like Desmos or GeoGebra to verify the answer.

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