To solve the given problem, we need to find [tex]\((f \div g)(x)\)[/tex], which means dividing the function [tex]\( f(x) \)[/tex] by the function [tex]\( g(x) \)[/tex].
Given the functions:
[tex]\[ f(x) = x^2 + 4x - 21 \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]
We need to determine:
[tex]\[ \left( \frac{f}{g} \right)(x) \][/tex]
Follow these steps:
1. Set Up the Division:
We need to divide [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex]:
[tex]\[
\left( \frac{x^2 + 4x - 21}{x - 3} \right)
\][/tex]
2. Perform Polynomial Division:
To simplify the expression, divide [tex]\( x^2 + 4x - 21 \)[/tex] by [tex]\( x - 3 \)[/tex]:
- First Term:
Divide the leading term of the numerator [tex]\( x^2 \)[/tex] by the leading term of the denominator [tex]\( x \)[/tex]:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
Multiply [tex]\( x \)[/tex] by the entire denominator [tex]\( x - 3 \)[/tex]:
[tex]\[
x \cdot (x - 3) = x^2 - 3x
\][/tex]
Subtract this product from the original numerator:
[tex]\[
(x^2 + 4x - 21) - (x^2 - 3x) = 4x - 3x - 21 = 7x - 21
\][/tex]
- Second Term:
Divide the new leading term [tex]\( 7x \)[/tex] by [tex]\( x \)[/tex]:
[tex]\[
\frac{7x}{x} = 7
\][/tex]
Multiply [tex]\( 7 \)[/tex] by the denominator [tex]\( x - 3 \)[/tex]:
[tex]\[
7 \cdot (x - 3) = 7x - 21
\][/tex]
Subtract this product from the new remainder:
[tex]\[
(7x - 21) - (7x - 21) = 0
\][/tex]
3. Combine the Quotients:
The division gives us:
[tex]\[
x + 7
\][/tex]
Thus, we rewrite the original fraction:
[tex]\[
\frac{x^2 + 4x - 21}{x - 3} = x + 7
\][/tex]
4. Express in Standard Form:
Finally, we express the result of [tex]\((f \div g)(x)\)[/tex] in a standard algebraic form:
[tex]\[
\left( \frac{f}{g} \right)(x) = x + 7 + \frac{0}{x - 3}
\][/tex]
Summarizing the entire division and steps, we obtain:
[tex]\[ \left( \frac{f}{g} \right)(x) = x + 7 \][/tex]
And therefore, the final answer is:
[tex]\[ \boxed{x + 7} \][/tex]
