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Factor trinomials mentally! Tips and tricks

Have you ever wondered how can we factor trinomials without writing anything on paper, i.e. mentally? The best way, unfortunately, is through trial-and-error method. However, to make factoring mentally faster, the trial-and-error must be an educated and systematic one.

Here are some tips and tricks in factoring the trinomial $ax^{2}+bx+c$ mentally. Once you master the techniques in this blog, you can simplify expressions and solve equations that require factoring with "lightning" speed, and impress your friends.

  • If $a+b+c=0$, then one of the factors is $(x-1)$. The other one is $(ax-c)$.
  • If $a+c=b$, then one of the factors is $(x+1)$. The other one is $(ax+c)$.
  • If $b$ and $c$ are both positive, then each factor takes a $+$ sign.
  • If $b$ is negative and $c$ is positive, then each factor takes a $-$ sign.
  • If $c$ is negative, then one of the factors take a $+$ sign, and the other takes a $-$ sign.
    • When selecting combinations of signs in this case, observe it from the resulting outer and inner terms, and the middle term of the original trinomial.
  • When selecting for combinations of $ax^{2}$, try to eliminate some combinations based on the parity of the middle terms (including the resulting outer and inner terms). Parity means whether a number is even or odd.
  • When selecting for combinations of $c$, try to eliminate some combinations based on the following.
    • If one of the factors is still factored by the GCF method, but the original trinomial is not factorable by the GCF method.
    • If either the resulting outer or inner term is too large compared to the middle term of the original polynomial.
  • Do not do trial-and-error check on all cases, only check ones that you think will yield a middle term close to the middle term of the original trinomial.
  • Repeatedly practice finding combinations mentally. This will help you find factors and combinations faster, and later on, you will be able to find factors without exerting too much thinking at all.

Example 1

Factor the trinomial $2x^{2}-x-6$.

Example 2

Factor the trinomial $4x^{2}-19x+21$.

Example 3

Factor the trinomial $45x^{2}+79x-124$.

Example 4

Factor the trinomial $10x^{2}-17xy-27y^{2}$.

Example 5

Factor the trinomial $12x^{2}+7x-12$.

Practice problems

Here are some more trinomials to practice what you have learned in this blog.

  1. $ 3x^2 - 13x - 10 $
  2. $ 40x^2 - 79x + 39 $
  3. $ 8x^2 + 26x + 15 $
  4. $ 16x^2 - 33x - 49 $
  5. $ 18x^2 + 9xy - 20y^2 $
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