Simplify: n^3 + 2n^2 - 36n - 72
Understand the Problem
The problem requires us to simplify the polynomial. We will start by finding common factors and then try to factorise if possible.
Answer
$(n+2)(n-6)(n+6)$
Answer for screen readers
$(n+2)(n-6)(n+6)$
Steps to Solve
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Group the terms Group the first two terms and the last two terms together. $$ (n^3 + 2n^2) + (-36n - 72) $$
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Factor out the greatest common factor (GCF) from each group
From the first group, the GCF is $n^2$. Factoring this out gives $n^2(n+2)$. From the second group, the GCF is $-36$. Factoring this out gives $-36(n+2)$.
$$ n^2(n+2) - 36(n+2) $$
- Factor out the common binomial factor
Notice that both terms now have a common factor of $(n+2)$. Factor this out: $$ (n+2)(n^2 - 36) $$
- Factor the difference of squares
Recognize that $n^2 - 36$ is a difference of squares. $n^2 - 36 = n^2 - 6^2 = (n-6)(n+6)$
- Write the final factored form
Substitute the factored form of $n^2 - 36$ back into the expression: $$ (n+2)(n-6)(n+6) $$
$(n+2)(n-6)(n+6)$
More Information
The polynomial $n^3 + 2n^2 - 36n - 72$ can be factored into $(n+2)(n-6)(n+6)$. This means that the roots of the polynomial are $n = -2$, $n = 6$, and $n = -6$.
Tips
A common mistake is to stop at the first factoring step $n^2(n+2) - 36(n+2)$ or $(n+2)(n^2 - 36)$. It is important to recognize $n^2-36$ as a difference of squares that can be further factored.
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