Question

factor completely. \\( f^2 - 8f + 7 \\)

Explanation:

Step1: Find two numbers that multiply to 7 and add to -8.

We need two numbers \(a\) and \(b\) such that \(a\times b = 7\) and \(a + b=-8\). The numbers are \(-1\) and \(-7\) since \((-1)\times(-7)=7\) and \((-1)+(-7)=-8\).

Step2: Rewrite the middle term using these numbers.

\(f^{2}-8f + 7=f^{2}-f-7f + 7\)

Step3: Group the terms and factor by grouping.

Group the first two terms and the last two terms: \((f^{2}-f)+(-7f + 7)\)
Factor out the greatest common factor from each group: \(f(f - 1)-7(f - 1)\)
Now, factor out the common binomial factor \((f - 1)\): \((f - 1)(f - 7)\)

Answer:

\((f - 1)(f - 7)\)