To find [tex]\((f \div g)(x)\)[/tex] given the functions [tex]\(f(x) = x^2 + 6x + 8\)[/tex] and [tex]\(g(x) = x + 4\)[/tex], we need to perform polynomial division. Let's perform this step-by-step.
1. Set up the division: We want to divide [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex]:
[tex]\[
\frac{x^2 + 6x + 8}{x + 4}
\][/tex]
2. Determine the first term of the quotient: Divide the leading term of the numerator by the leading term of the denominator. The leading term of the numerator [tex]\(x^2\)[/tex] divided by the leading term of the denominator [tex]\(x\)[/tex] is [tex]\(x\)[/tex]. So, the first term in the quotient is [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply [tex]\(x\)[/tex] by [tex]\((x + 4)\)[/tex] and subtract this from the original polynomial [tex]\(f(x)\)[/tex].
[tex]\[
x^2 + 6x + 8 - x(x + 4) = x^2 + 6x + 8 - x^2 - 4x = 2x + 8
\][/tex]
4. Determine the next term of the quotient: Now, divide the leading term of the remaining polynomial by the leading term of the denominator. The leading term of [tex]\(2x\)[/tex] divided by the leading term of [tex]\(x\)[/tex] is [tex]\(2\)[/tex]. So, the next term in the quotient is [tex]\(2\)[/tex].
5. Multiply and subtract: Multiply [tex]\(2\)[/tex] by [tex]\((x + 4)\)[/tex] and subtract this from the remaining polynomial.
[tex]\[
2x + 8 - 2(x + 4) = 2x + 8 - 2x - 8 = 0
\][/tex]
6. Combine the quotient terms: Adding up the terms we found for the quotient, we have:
[tex]\[
(f \div g)(x) = x + 2
\][/tex]
So, the division of [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex] yields a quotient of [tex]\(x + 2\)[/tex].
Since the remainder is [tex]\(0\)[/tex], we can express [tex]\((f \div g)(x)\)[/tex] as:
[tex]\[
(f \div g)(x) = x + 2
\][/tex]
Thus, the polynomial form of [tex]\((f \div g)(x)\)[/tex] in the simplest form is:
[tex]\[
(f \div g)(x) = x + 2
\][/tex]
