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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 that can be factored into the square of a binomial. It has the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). When factoring, the trinomial \(x^2 + 2xy + y^2\) becomes \((x + y)^2\) and the trinomial \(x^2 - 2xy + y^2\) becomes \((x - y)^2\).

Step by step solution

01

Define Perfect Square Trinomial

A Perfect Square Trinomial is a trinomial that can be factored into the square of a binomial. It takes the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). In this form, \(a\) and \(b\) are real numbers and \(a\), being the coefficient of the squared first term, is a positive real number.
02

Identify Perfect Square Trinomial

To identify if a trinomial is a perfect square, it must fit one of the two forms mentioned. If the trinomial is \(x^2 + 2xy + y^2\) or \(x^2 - 2xy + y^2\), then \(a = x\) and \(b = y\) or \(b = -y\). Check if the middle term is twice the product of the square roots of the first and third term.
03

Factor Perfect Square Trinomial

After successfully identifying a perfect square trinomial, you can factor it into the square of a binomial. If the trinomial is in the form \(x^2 + 2xy + y^2\), it is factored as \((x + y)^2\), and if it is in the form \(x^2 - 2xy + y^2\), it is factored as \((x - y)^2\).

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Most popular questions from this chapter

Simplify using properties of exponents. $$\left(x^{\frac{4}{5}}\right)^{5}$$

Simplify using properties of exponents. $$\left(x^{\frac{2}{3}}\right)^{3}$$

a. Simplify: \(21 x+10 x\) b. Simplify: \(21 \sqrt{2}+10 \sqrt{2}\)

Place the correct symbol, \(>\) or \(<,\) in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. $$\text { a. } 3^{\frac{1}{2}} \square 3^{\frac{1}{3}}$$ $$\text { b. } \sqrt{7}+\sqrt{18} \square \sqrt{7}+18$$

Why must \(a\) and \(b\) represent non negative numbers when we write \(\sqrt{a} \cdot \sqrt{b}=\sqrt{a b ?}\) Is it necessary to use this restriction in the case of \(\sqrt[3]{a} \cdot \sqrt[3]{b}=\sqrt[3]{a b} ?\) Explain.
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