Question

squaring a binomial
louise completed the work shown below.
\\((5x^3 + 3)^2 = (5x^3)^2 + (3)^2 = 25x^6 + 9\\)
determine if louise’s answer is correct. explain.

Explanation:

Step1: Recall the formula for squaring a binomial

The formula for squaring a binomial \((a + b)^2\) is \(a^2 + 2ab + b^2\), not \(a^2 + b^2\). Here, \(a = 5x^3\) and \(b = 3\).

Step2: Apply the correct formula to \((5x^3 + 3)^2\)

Using the formula \((a + b)^2=a^2 + 2ab + b^2\), we substitute \(a = 5x^3\) and \(b = 3\).
First, calculate \(a^2=(5x^3)^2 = 25x^6\), \(b^2 = 3^2=9\), and \(2ab=2\times(5x^3)\times3=30x^3\).
Then, \((5x^3 + 3)^2=(5x^3)^2+2\times(5x^3)\times3+(3)^2=25x^6 + 30x^3+9\).
Louise's answer is \(25x^6 + 9\), which is missing the middle term \(30x^3\). So Louise's answer is incorrect.

Answer:

Louise's answer is incorrect. The formula for squaring a binomial \((a + b)^2\) is \(a^2+2ab + b^2\), not \(a^2 + b^2\). For \((5x^3 + 3)^2\), using the correct formula, we get \((5x^3)^2+2\times(5x^3)\times3+(3)^2 = 25x^6+30x^3 + 9\), while Louise's answer is \(25x^6 + 9\) (missing the \(30x^3\) term).