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Chapter 0: Problem 127

What is a perfect square trinomial and how is it factored?

Short Answer

Expert verified
A perfect square trinomial is a trinomial of the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\) and it is factored as \((a+b)^2\) or \((a-b)^2\), respectively.

Step by step solution

01

Define Perfect Square Trinomial

A perfect square trinomial is a type of trinomial which can be factored into a binomial squared. It has the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). The first and third terms are square numbers, and the middle term is twice the product of square roots of the first and third terms.
02

Identify Perfect Square Trinomial

To identify a perfect square trinomial, check if the first and third terms are square numbers and if the middle term is twice the product of their square roots. If it matches either form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\), then it is a perfect square trinomial.
03

Factor the Perfect Square Trinomial

To factor a perfect square trinomial, find the square roots of the first and third terms. The binomial is then the square root of the first term followed by the sign of the middle term and then the square root of the third term. The perfect square trinomial \(a^2 + 2ab + b^2\) is factored as \((a+b)^2\), and \(a^2 - 2ab + b^2\) is factored as \((a-b)^2\). Both factored forms are squared because the trinomial is a perfect square.

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Most popular questions from this chapter

Factor completely, or state that the polynomial is prime. $$5 x^{3}-45 x$$

List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. $$\left\\{-11,-\frac{5}{6}, 0,0.75, \sqrt{5}, \pi, \sqrt{64}\right\\}$$

Factor each perfect square trinomial. $$4 x^{2}+4 x+1$$

Factor each trinomial, or state that the trinomial is prime. $$6 x^{2}-7 x y-5 y^{2}$$

Simplify the radical expressions in Exercises \(67-74\) if possible. $$\sqrt[3]{150}$$
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