Chapter 5.8, Problem 17WE

To determine

To factor : the given expression

Answer to Problem 17WE

The expression $a^2-ab-42b^2$ is $\boxed{\left(a-7b\right)\left(a+6b\right)}$

Explanation of Solution

Given information : $a^2-ab-42b^2$

Concept Involved:

Whenever we have a trinomial of the form $p{x}^2+qx+r$ , follow the below steps to factor it pr or grouping method

1. Identify the SUM { q } and PRODUCT { pr } of the expression
2. Find two factors two rewrite the middle term m and n in such a way that it is multiplied to give PRODUCT and added to give the SUM of the expression.
{Hint: List the factor pair of pr and see which two factors add up to give q to get the numbers m and n .}
3. Use the two numbers m and n to rewrite the middle term to look as $p{x}^2+mx+nx+r$
4. Factor the GCF of first two terms and factor the GCF of last two terms. {Note: we will end up getting common binomial after this step}
5. Factor the common binomial from the whole expression to get the final factored expression.

Calculation:

Identify the a, b and c of the given expression $a^2-ab-42b^2$

  $p=1;q=-b;r=-42b^2$

Identify the SUM { q } and PRODUCT { pr } of the expression

  $SUM=-b;$

  $PRODUCT=-42b^2$

List the factor pairs of $-42b^2$ and see which two pair add up to give - b .

Factor pairs of $-42b^2$ are $\left\{-1b,42b\right\},\left\{1b,-42b\right\},\left\{2b,-21b\right\},\left\{-2b,21b\right\},\left\{-3b,14b\right\},$ $\left\{3b,-14b\right\},\left\{-6b,7b\right\},\&\left\{6b,-7b\right\}$ and the pair $\left\{6b,-7b\right\}$ add up to - b

Use the - 7b & 6b to rewrite the middle term of the original expression

  $a^2-7ab+6ab-42b^2$

Factor GCF $a$ from first two terms and $6b$ from last two terms

  $a\left(a-7b\right)+6b\left(a-7b\right)$

Factor the common binomial $a-7b$ from the whole expression

  $\left(a-7b\right)\left(a+6b\right)$

Check:

Multiplying first outer inner last terms

  $a^2+6ab-7ab-42b^2$

Combine like terms and see whether we get the original expression

  $a^2-ab-42b^2$ , Yes! that’s the original expression

Conclusion:

The expression $a^2-ab-42b^2$ factored to get is $\boxed{\left(a-7b\right)\left(a+6b\right)}$ .