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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 is the square of a binomial. It is either in the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). It is factored by identifying the suitable 鈥榓鈥� and 鈥榖鈥� values (where a and b are numbers or terms) in the trinomial and putting them into \( (a+b)^2\) or \((a-b)^2\), depending on the sign of the 2ab term.

Step by step solution

01

Understanding Perfect Square Trinomial

A perfect square trinomial is formed when a binomial is multiplied by itself. When an equation \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\) is given, it is recognized as a perfect square trinomial. In the first case, when factored, it will give \((a+b)^2\), and in the second case, it will give \((a-b)^2\).
02

Factoring Perfect Square Trinomial

To factor this kind of trinomial, first, ensure it is in the format described above, either \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). Then identify the values of a and b. For example, for the trinomial \(x^2 + 10x + 25\), when it is in the form of a perfect square trinomial, it gives a = x and b = 5. To confirm whether it's a perfect square trinomial, note that the middle term is equivalent to 2ab which in this case coincides with 2*x*5 = 10x. To factor, simply write it as \((a+b)^2\) or \((a-b)^2\), depending on whether 2ab is positive or negative. Here, the factored form would be \((x+5)^2\).
03

Practice with Other Examples

To fully master this, try with other examples by spotting a pattern between the coefficients and working out the binomial. For instance, in solving \(12y^2 + 32y + 16\), here, a = 2y, b = 4, plug into the formula and the factored form is \(2y + 4)^2\).

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Most popular questions from this chapter

Explain how to solve \(x^{2}+6 x+8=0\) using the quadratic formula.

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