Chapter 8.7, Problem 42P

Factor each expression.

$42.45t^2-72t+24$

To determine

To factor : the given expression

Answer to Problem 42P

Factoring the expression $45t^2-72t+24$ we get $\boxed{3\left(15t^2-24t+8\right)}$

Explanation of Solution

Given information : $45t^2-72t+24$

Concept Involved:

Whenever we have a trinomial of the form $a{x}^2+bx+c$ , follow the below steps to factor it ac or grouping method

1. Identify the SUM { b } and PRODUCT { ac } 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 ac and see which two factors add up to give b to get the numbers m and n .}

3. Use the two numbers m and n to rewrite the middle term to look as $a{x}^2+mx+nx+c$
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 $45t^2-72t+24$

$a=45;b=-72;c=24$

Identify the SUM { b } and PRODUCT { ac } of the expression

$SUM=-72;$

$PRODUCT=1080$

List the factor pairs of $1080$ .

Factor pairs of $1080$ are $\left\{1,1080\right\},\left\{-1,-1080\right\},\left\{2,540\right\},\left\{-2,-540\right\},\left\{3,360\right\},\left\{-3,-360\right\},$

$\left\{4,270\right\},\left\{-4,-270\right\},\left\{5,216\right\},\left\{-5,-216\right\},\left\{6,180\right\},\left\{-6,-180\right\},\left\{8,135\right\},\left\{-8,-135\right\},\left\{9,120\right\},\left\{-9,-120\right\}$

$\left\{10,108\right\},\left\{-10,-108\right\}$ , $\left\{12,90\right\},\left\{-12,-90\right\}$ , $\left\{15,72\right\},\left\{-15,-72\right\}$ , $\left\{18,60\right\},\left\{-18,-60\right\},$ $\left\{20,54\right\},$ $\left\{-20,-54\right\},$ $\left\{24,45\right\},$ $\left\{-24,-45\right\},$ $\left\{30,36\right\},$ $\&\left\{-30,-36\right\}$ and there is no such factor pair that add up to -72, so trinomial $45t^2-72t+24$ is prime.

Factor the GCF 3 from the whole expression

$3\left(15t^2-24t+8\right)$

Conclusion:

We can factor the GCF 3 from the whole expression $45t^2-72t+24$ to get $3\left(15t^2-24t+8\right)$ and cannot be factored further because $15t^2-24t+8$ is PRIME.