Chapter 8, Problem 43SGR

To determine

To find: The factor of each trinomial, if trinomial cannot be factored find the prime.

Answer to Problem 43SGR

The factor of each trinomial is: $\left[(x+3)(3x-15)\right]\left[(x+4)(x+5)\right]$

Explanation of Solution

Given information:

The given expression is $3x^2-6x-45$

Calculation:

Considering the expression:

The objective is to factor the given expression $3x^2-6x-45$

The product of $3x^2$ term and constant term is

  $3x^2\times (-45)=-135x^2$

The middle term of the trinomial is $-6x$ .

The expression can be factored as below:-

Consider $3x^2+mx+px-45$ be the factor of the trinomial $3x^2-6x-45$ .

The coefficient of product of $x^2$ term and constant term is negative and coefficient of middle term of the polynomial is negative. Thus, the one value is positive and lesser than second.

Calculate the value of $m$ and $p$ as,

  $\begin{array}{cc}\begin{array}{c}\text{Factors}\\ \text{of 135}\end{array}& \begin{array}{l}\text{Sum of}\hfill \\ \text{Factors}\hfill \end{array}\\ 1,-135& -134\\ 3,-45& -42\\ 5,-27& -22\\ 9,-15& -6\end{array}$

  $3x^2-6x-45=3x^2-15x+9x-45$ The correct factors are $9$ and $-15$

  $=(x+3)(3x-15)$

Therefore, the factor of the trinomial are $\left[(x+3)(3x-15)\right]\left[(x+4)(x+5)\right]$