Problem 93 - Introductory Algebra for College Students

Question

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$16 a^{2}-32 a b+12 b^{2}$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(16 a^{2}-32 a b+12 b^{2}\) is \[4 (2a - \sqrt{3}b)^2\].

Step 1: Identify Common Factors

The given polynomial is \(16 a^{2}-32 a b+12 b^{2}\). Perform the first step of identifying common factors among all terms. We can see that '4' is the common factor among all terms.

Step 2: Factor out the Greatest Common Factor

When the greatest common factor '4' is factored out, the equation turns into: \[4(4 a^{2}-8 a b+3 b^{2})\].

Step 3: Apply the Perfect Square Trinomial Formula

The perfect square trinomial equation form is \(a^{2}-2ab+b^{2}\). With closer examination, the expression inside the brackets matches with this formula, where \(a=2a\) and \(b=\sqrt{3}b\).

Step 4: Put in the Form of Perfect Squares

Putting the equation in the form of two perfect squares, the equation transforms to: \[4 (2 a - \sqrt{3} b)^2\].

Step 5: Expanding to Check

Expand the factored polynomial to check the result, \(4 (2 a - \sqrt{3} b) *(2 a - \sqrt{3} b) = 16 a^{2}-32 a b+12 b^{2}\), which confirms the factored form.