Chapter 4.3, Problem 84HP

To determine

To express: The factors polynomial of the form $a{x}^2+bx+c$ with the help of guess-and-check strategy.

Answer to Problem 84HP

One of the polynomial can be $2x^2+5x-3$ .

Explanation of Solution

Given information:

The polynomial is factored with help of guess-and-check strategy.

Formula used:

To factor a polynomial split the middle term in such a way that sum of two numbers is the middle term and product of two numbers is same is product of first and last term.

Calculation:

Consider the polynomial of the form $a{x}^2+bx+c$ say $2x^2+5x-3$ .

Recall that to factor a polynomial split the middle term in such a way that sum of two numbers is the middle term and product of two numbers is same is product of first and last term.

For the above polynomial $-1$ and $6$ are two numbers such that their sum is $5$ and product is $-6$ .

Apply it, first the group the similar terms,

  $\begin{array}{c}2x^2+5x-3=2x^2-x+6x-3\\ =x\left(2x-1\right)+3\left(2x-1\right)\\ =\left(2x-1\right)\left(x+3\right)\end{array}$

Thus, the polynomial is factored as $\left(2x-1\right)\left(x+3\right)$ .