Question

factor the following polynomial.\\(x^2 - 8x + 16\\)\\((x - ?)^2\\)

Explanation:

Step1: Recall the perfect square formula

The perfect square trinomial formula is \(a^2 - 2ab + b^2=(a - b)^2\). For the polynomial \(x^2-8x + 16\), we can compare it with the form \(a^2-2ab + b^2\). Here, \(a = x\), and \(2ab=8x\). Since \(a=x\), we substitute \(a\) into \(2ab = 8x\), getting \(2\times x\times b=8x\).

Step2: Solve for \(b\)

Divide both sides of \(2xb = 8x\) (assuming \(x
eq0\); if \(x = 0\), the equation still holds for the value we find) by \(2x\). So \(\frac{2xb}{2x}=\frac{8x}{2x}\), which simplifies to \(b = 4\). We can also check by squaring \((x - 4)\): \((x - 4)^2=x^2-8x + 16\), which matches the given polynomial.

Answer:

\(4\)