Question

topic: finding the roots and factors of a polynomial
use the given root to find the remaining roots. then write the function in factored form.
14.
$f(x) = x^3 - x^2 - 17x - 15$
$x = -1$
15.
$f(x) = x^3 - 3x^2 - 61x + 63$
$x = 1$
16.
$f(x) = 6x^3 - 18x^2 - 60x$
$x = 0$
17.
$f(x) = x^3 - 14x^2 + 57x - 72$
$x = 8$

1. a relationship exists between the roots of a function and the constant term of the function. look back at the roots and the constant term in each problem. make a statement about anything you notice.

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Explanation:

Step1: Factor out \(x\) from the polynomial

Given \(f(x) = 6x^{3}-18x^{2}-60x\), we can factor out the greatest common factor, which is \(6x\). So we have \(f(x)=6x(x^{2}-3x - 10)\).

Step2: Factor the quadratic expression

Now we need to factor the quadratic \(x^{2}-3x - 10\). We look for two numbers that multiply to \(- 10\) and add up to \(-3\). The numbers are \(-5\) and \(2\) since \((-5)\times2=-10\) and \(-5 + 2=-3\). So \(x^{2}-3x - 10=(x - 5)(x+2)\).

Step3: Write the factored form and find roots

Putting it all together, the factored form of \(f(x)\) is \(f(x)=6x(x - 5)(x + 2)\). To find the roots, we set \(f(x)=0\). Using the zero - product property, if \(ab = 0\) then either \(a = 0\) or \(b = 0\). We know one root is \(x = 0\) (given). For \(x-5=0\), we get \(x = 5\); for \(x + 2=0\), we get \(x=-2\).

Answer:

The remaining roots are \(x = 5\) and \(x=-2\), and the factored form of \(f(x)\) is \(f(x)=6x(x - 5)(x + 2)\)