Question

question
factor to find all x-intercepts of the function.
$f(x) = x^3 + 3x^2 - 9x - 27$
answer attempt 1 out of 2
+ additional solution - no solution
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Explanation:

Step1: Group terms

Group the first two terms and the last two terms:
$$f(x)=(x^3 + 3x^2)+(-9x - 27)$$

Step2: Factor out GCF from each group

Factor out \(x^2\) from the first group and \(-9\) from the second group:
$$f(x)=x^2(x + 3)-9(x + 3)$$

Step3: Factor out common binomial

Factor out \((x + 3)\) from both terms:
$$f(x)=(x + 3)(x^2 - 9)$$

Step4: Factor difference of squares

Notice that \(x^2 - 9\) is a difference of squares (\(a^2 - b^2=(a + b)(a - b)\)) with \(a = x\) and \(b = 3\):
$$f(x)=(x + 3)(x + 3)(x - 3)=(x + 3)^2(x - 3)$$

Step5: Find x-intercepts

To find the x-intercepts, set \(f(x)=0\):
$$(x + 3)^2(x - 3)=0$$
Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\).

• If \((x + 3)^2=0\), then \(x=-3\) (with multiplicity 2).
• If \(x - 3=0\), then \(x = 3\).

Answer:

The x - intercepts are \(x=-3\) (with multiplicity 2) and \(x = 3\). In terms of the points on the x - axis, the x - intercepts are \((-3,0)\) and \((3,0)\).