Question

warm-up
use factoring to rewrite this polynomial expression.
drag each factor to the correct location on the equation. each factor can be used more than once, but not all factors will be used.
$x^3 + 5x^2 - 9x - 45 = (\quad)(\quad)(\quad)$
$x - 3$ $x - 5$ $x + 3$ $x + 5$ $x + 9$ $x - 9$
$x + 1$ $x - 1$

Explanation:

Step1: Group the terms

We group the first two terms and the last two terms of the polynomial:
$$(x^3 + 5x^2) + (-9x - 45)$$

Step2: Factor out the GCF from each group

For the first group \(x^3 + 5x^2\), the greatest common factor (GCF) is \(x^2\), so we factor it out: \(x^2(x + 5)\)
For the second group \(-9x - 45\), the GCF is \(-9\), so we factor it out: \(-9(x + 5)\)
Now the polynomial becomes:
$$x^2(x + 5)-9(x + 5)$$

Step3: Factor out the common binomial factor

We can see that \((x + 5)\) is a common factor, so we factor it out:
$$(x + 5)(x^2 - 9)$$

Step4: Factor the difference of squares

Notice that \(x^2 - 9\) is a difference of squares, which can be factored as \((x + 3)(x - 3)\) (since \(a^2 - b^2=(a + b)(a - b)\) where \(a = x\) and \(b = 3\))
So now we have:
$$(x + 5)(x + 3)(x - 3)$$

Answer:

\((x + 5)(x + 3)(x - 3)\)