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Chapter 6: Problem 112

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

Short Answer

Expert verified
A perfect square trinomial is a specific type of trinomial that fits the formula \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). It is factored to \((a+b)^2\) and \((a-b)^2\) respectively, where 'a' and 'b' are the square roots of the first and third terms of the trinomial.

Step by step solution

01

Recognize the Pattern in the Trinomial

A perfect square trinomial fits one of two formulas: \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). It's important to recognize this pattern to identify if the given trinomial is in fact a perfect square trinomial.
02

Determine the Terms a and b

Once the pattern is recognized, determine the values of a and b. 'a' and 'b' represent the two terms in the binomial that would be squared to get the trinomial. 'a' is the square root of the first term in the trinomial and 'b' is the square root of the third term.
03

Confirm Your Identification of a and b

Check your identification of a and b by multiplying them together, doubling the result, and checking if it matches the second term of the trinomial. If it does, you鈥檝e correctly identified a and b.
04

Factor the Trinomial

Now, knowing the values of 'a' and 'b', the perfect square trinomial can be factored. If the trinomial fits the form \(a^2 + 2ab + b^2\), it can be factored to \((a+b)^2\). If the trinomial fits the form \(a^2 - 2ab + b^2\), then it factors to \((a-b)^2\).

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Most popular questions from this chapter

The formula $$S=2 x^{2}-12 x+82$$ models spending by international travelers to the United States, \(S,\) in billions of dollars, \(x\) years after \(2000 .\) Use this formula to solve. In which years did international travelers spend \(\$ 72\) billion?

Graph: \(y=-\frac{2}{3} x+1 .\) (Section 3.4, Example 3)

In Exercises \(124-127,\) factor each polynomial. $$x^{2 n}-25 y^{2 n}$$

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-12 x+9=(4 x-3)^{2} ;[-5,5,1] \text { by }[0,20,1]$$

In Exercises 142-146, use the GRAPH or TABLE feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. \(2 x^{3}+10 x^{2}-2 x-10=2(x+5)\left(x^{2}+1\right) ;[-8,4,1]\) by [-100,100,10]
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