Chapter 7, Problem 7CT

To determine

To explain: Explain how to determine whether a polynomial is a perfect square trinomial.

Answer to Problem 7CT

The first term has to be a perfect square.

The third term should also be a perfect square.

The middle term should be either  $2$  or  $-2$  times the product of the square root of the first term and the square root of the third term.

Explanation of Solution

Given information: The polynomial to be checked is a trinomial.

Calculation:

General form of a perfect square trinomial is $a^2{x}^2\pm 2axb+b^2$

It can be xpressed in squared-binomial form as ${(ax\pm b)}^2$

A trinomial is a perfect square if it satisfies the following conditions:

The first term has to be a perfect square.

The third term should also be a perfect square.

The middle term should be either  $2$  or  $-2$  times the product of the square root of the first term and the square root of the third term.

Example:-

  $x^2+10x+25$

Square root of first term is $x$ and that of the third term is $5$ .

  $10x=x(5)(2)$

  $\therefore x^2+10x+25$ is a perfect square trinomial and can be represented in squared binomial form as ${(x+5)}^2$ .