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Chapter 0: Problem 36

Factor each trinomial, or state that the trinomial is prime. $$3 x^{2}+4 x y+y^{2}$$

Short Answer

Expert verified
The factorized form of \(3x^2 +4xy + y^2\) is \((\sqrt{3}x + y)^2\)

Step by step solution

01

Identify the form

Observe the form of the trinomial. It resembles the quadratic form \(a^2 x^2 + 2abxy + b^2 y^2\) which is a form of a perfect square trinomial \((ax + by)^2\). Here, we need to identify the values of \(a\), \(b\), \(x\), and \(y\) in order to factor.
02

Compare the given trinomial.

By comparing \(3x^2 +4xy + y^2\) to the form \(a^2 x^2 + 2abxy + b^2 y^2\). We can deduce the following: \(a = \sqrt{3}\), \(b = 1\), \(x = x\), \(y = y\) since \(\sqrt{3}^2 = 3\), \(1^2 = y^2\), \(2*\sqrt{3}*1 = 4\) and \(x = x\), \(y = y\)
03

Factorize the trinomial.

Now putting these values into the binomial form \((ax+by)^2\), the trinomial can be factorised to: \((\sqrt{3}x + y)^2\)

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Most popular questions from this chapter

Why must \(a\) and \(b\) represent nonnegative 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.

Factor and simplify each algebraic expression. $$\left(x^{2}+3\right)^{-\frac{1}{3}}+\left(x^{2}+3\right)^{-\frac{1}{3}}$$

In parts (a) and (b), complete each statement. $$\text { a. } b^{4} \cdot b^{3}=(b \cdot b \cdot b \cdot b)(b \cdot b \cdot b)=b^{?}$$ $$\text { b. } b^{5} \cdot b^{5}=(b \cdot b \cdot b \cdot b \cdot b)(b \cdot b \cdot b \cdot b \cdot b)=b^{7}$$ c. Generalizing from parts (a) and (b), what should be done with the exponents when multiplying exponential expressions with the same base?

Describe what it means to rationalize a denominator. Use both \(\frac{1}{\sqrt{5}}\) and \(\frac{1}{5+\sqrt{5}}\) in your explanation.

Will help you prepare for the material covered in the next section. A. Use a calculator to approximate \(\sqrt{300}\) to two decimal places. B. Use a calculator to approximate \(10 \sqrt{3}\) to two decimal places. C. Based on your answers to parts (a) and (b), what can you conclude?
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