Q. 357 - Intermediate Algebra

Question

In the following exercises, factor each trinomial of the form $x^2+bx+c$

$m^2+3m-54$

Step-by-Step Solution

Answer

The factored form of the polynomial is,

$\left(m+9\right)\left(m-6\right)$

Step 1. Given Information

We are given a polynomial,  

$m^2+3m-54$

Step 2. Factorizing the polynomial

Using the product of $ac$ method in factorizing the polynomial of form $a{x}^2+bx+c$, we get

$$\begin{gathered} 9\times 6=54 \\ 9-6=54 \end{gathered}$$

So, Splitting the middle term, 

$m^2+3m-54=m^2+9m-6m-54$

Grouping terms, we get 

$m^2+3m-54=\left(m^2+9m\right)-\left(6m+54\right)$

Factoring out GCF from groups, we get 

$m^2+3m-54=m\left(m+9\right)-6\left(m+9\right)$

Again, factoring out GCF, we get 

$m^2+3m-54=\left(m+9\right)\left(m-6\right)$

Step 3. Checking the factorization

Multiplying the factors,  

$$\begin{gathered} \left(m+9\right)\left(m-6\right)=m^2+3m-54 \\ m\left(m-6\right)+9\left(m-6\right)=m^2+3m-54 \\ {m}^2-6m+9m-54=m^2+3m-54 \\ {m}^2+3m-54=m^2+3m-54 \\ LHS=RHS \end{gathered}$$

Hence the factorization is correct.