Problem 92 - Elementary Algebra

Question

(a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Use a pattern to factor \(9 x^{2}-6 x+1\). Incorrect Answer: Since the pattern is \(a^{2}+2 a b+b^{2}=\) \((a+b)(a+b), a=3 x\) and \(b=1\), and the factored polynomial is \((3 x+1)(3 x+1)\).

Step-by-Step Solution

Answer

\((9x^2 - 6x + 1) = (3x - 1)^2\)

Step 1: Identify the mistake

The incorrect answer states, 'Since the pattern is \(a^{2}+2ab+b^{2} = (a+b)(a+b)\),' and gives \(a = 3x\) and \(b = 1\). The mistake here is in the identification of the pattern. The correct pattern should be compared with \( (a-b)^{2} = a^2 - 2ab + b^2 \).

Step 2: Recognize the correct pattern

The given polynomial is \(9x^2 - 6x + 1\). This can be compared to \(a^2 - 2ab + b^2 = (a-b)^2\).

Step 3: Determine a and b

Here, identify that \(a^2 = 9x^2\), so \(a = 3x\), and \(b^2 = 1\), so \(b = 1\).

Step 4: Factor using the correct pattern

Since the middle term -6x can be written as \(-2ab \) (i.e., \(-2 \times 3x \times 1\)), the polynomial can be factored as: \[9x^2 - 6x + 1 = (3x - 1)^2\].

Key Concepts

factoring by squares

Factoring by squares is an essential method to simplify quadratic polynomials. It involves expressing a quadratic expression in the form of a squared binomial. For instance, if we have an expression of the form \(a^2 - 2ab + b^2\), it can be rewritten as \((a - b)^2\).