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This equation appears in a question book but the method for factoring (if there is one) is not familiar to me.

$$x^3 + x^2 -4x -4 $$

The solution manual lists the following as the factorised form of the above equation:

$$x^2(x + 1) - 4(x + 1)$$

What is the method for achieving this? Thanks

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3 Answers 3

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This is just factoring out the common factor of $x^2$ from the first two terms, and the common factor of $-4$ from the last two terms. However, the correct factorization would be

$$x^2(x+1) - 4(x+1)$$

in that light. This suggests the notion of "factoring by grouping", if this is a keyword you want to look up further, since the next most reasonable factorization is factoring out the common factor of $x+1$:

$$(x^2-4)(x+1)$$

Khan Academy and this site discuss some examples in detail.


Note 1: The original question statement used the factorization of $x^2(x+1) - 4(x-1)$, hence my comment on the factorization being wrong. The question has since been edited.

Note 2: As astutely brought up in the comments, the $x^2-4$ factor can be further factored as $(x-2)(x+2)$. This is often known as the "difference of squares" identity. Further discussion and applications are on Khan Academy, Brilliant, and Wikipedia.

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    $\begingroup$ It was nice of you to add references. $\endgroup$
    – NoChance
    Feb 24, 2023 at 7:00
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    $\begingroup$ Worth mentioning that your first factor can be further broken into (x+2)(x-2) $\endgroup$ Feb 24, 2023 at 20:25
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Observe that for $p(x) = x^3+x^2 - 4x - 4$, then $p(-1) = 0$. Thus as a polynomial, $p(x)$ has a factor $x+1$. You can then use synthetic division to obtain the quotient and can factor $p(x)$ into a product of $x+1$ and a quadratic polynomial.

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  • $\begingroup$ Thanks for this good reminder of my student days! We learned lots of cool systematic methods to factor polynomials, but then during exams when time was precious and the clock was running, we never took the time to apply those methods. Instead we just guessed roots. Here the roots are -1, 2, -2. $\endgroup$
    – Stef
    Feb 24, 2023 at 19:18
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$-4(x+1)$, you mean. You then factor out the $(x+1)$, obtaining $(x^2-4)(x+1)$. This is often called factoring by grouping.

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