Question

given that $f(x) = x^2 - 11x + 30$ and $g(x) = x - 5$, find $(f - g)(x)$ and express the result as a polynomial in simplest form.

Explanation:

Step1: Recall the definition of \((f - g)(x)\)

By the definition of function subtraction, \((f - g)(x)=f(x)-g(x)\).

Step2: Substitute the given functions

We know that \(f(x)=x^{2}-11x + 30\) and \(g(x)=x - 5\), so substitute these into the formula:
\((f - g)(x)=(x^{2}-11x + 30)-(x - 5)\)

Step3: Distribute the negative sign

Distribute the negative sign to each term in \(g(x)\):
\((f - g)(x)=x^{2}-11x + 30 - x + 5\)

Step4: Combine like - terms

Combine the \(x\) - terms and the constant terms:
For the \(x\) - terms: \(-11x-x=-12x\)
For the constant terms: \(30 + 5 = 35\)
So, \((f - g)(x)=x^{2}-12x + 35\)

Answer:

\(x^{2}-12x + 35\)