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Chapter 15: Problem 385

Factor A) \(a^{4}+4 a^{2}+4\) B) \(9 a^{2}-6 a b^{2}+b^{4}\)

Short Answer

Expert verified
The factored forms of the given expressions are: A) \(a^4 + 4a^2 + 4 = (a^2 + 2)^2\) B) \(9a^2 - 6ab^2 + b^4 = (3a - b^2)^2\)

Step by step solution

01

Identify a Perfect Square Trinomial

Since all the coefficients are perfect squares and the exponent of the first term is double that of the middle term, we can suspect that this expression might be a perfect square trinomial. A perfect square trinomial is of the form (x + y)^2 = x^2 + 2xy + y^2.
02

Factor A

Let's check if it satisfies the perfect square trinomial condition: Here, x = \(a^2\) and y = 2 Now, \(x^2 = (a^2)^2 = a^4\), \(2xy = 2(a^2)(2) = 4a^2\), and \(y^2 = (2)^2 = 4\). These are the terms in our expression, so \(a^4 + 4a^2 + 4\) can be factored as: \((a^2 + 2)^2\). For B) \(9a^2 - 6ab^2 + b^4\):
03

Identify a Perfect Square Trinomial

Again, as all the coefficients are perfect squares and the exponent of the last term is double that of the middle term, this expression might also be a perfect square trinomial.
04

Factor B

Let's check if it satisfies the perfect square trinomial condition: Here, x = 3a and y = b^2 Now, \(x^2 = (3a)^2 = 9a^2\), \(2xy = 2(3a)(b^2) = 6ab^2\), and \(y^2 = (b^2)^2 = b^4\). These are the terms in our expression, but notice that the sign in front of the middle term is negative, so \(9a^2 - 6ab^2 + b^4\) can be factored as: \((3a - b^2)^2\).

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