To find [tex]\( f(x) \div g(x) \)[/tex] where [tex]\( f(x) = x^2 + 4x - 45 \)[/tex] and [tex]\( g(x) = x + 9 \)[/tex], we need to perform polynomial long division. Let's solve this step-by-step:
1. Set up the division:
We are dividing [tex]\( f(x) = x^2 + 4x - 45 \)[/tex] by [tex]\( g(x) = x + 9 \)[/tex].
2. Divide the leading terms:
The leading term of [tex]\( f(x) \)[/tex] is [tex]\( x^2 \)[/tex], and the leading term of [tex]\( g(x) \)[/tex] is [tex]\( x \)[/tex].
Divide [tex]\( x^2 \)[/tex] by [tex]\( x \)[/tex] to get the first term of the quotient: [tex]\( \frac{x^2}{x} = x \)[/tex].
3. Multiply and subtract:
Multiply [tex]\( x + 9 \)[/tex] by the first term of the quotient, which is [tex]\( x \)[/tex]:
[tex]\[ (x + 9) \cdot x = x^2 + 9x \][/tex]
Subtract this product from [tex]\( f(x) \)[/tex]:
[tex]\[
(x^2 + 4x - 45) - (x^2 + 9x) = 4x - 9x - 45 = -5x - 45
\][/tex]
4. Repeat the process:
Now, divide the new leading term [tex]\(-5x\)[/tex] by the leading term of [tex]\( g(x) \)[/tex], [tex]\( x \)[/tex]:
[tex]\[ \frac{-5x}{x} = -5 \][/tex]
Multiply [tex]\( x + 9 \)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[ (x + 9) \cdot (-5) = -5x - 45 \][/tex]
Subtract this product from [tex]\(-5x - 45\)[/tex]:
[tex]\[
(-5x - 45) - (-5x - 45) = 0
\][/tex]
5. Combine the results:
The quotient is [tex]\( x - 5 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
Thus, the result of [tex]\( f(x) \div g(x) \)[/tex] is:
[tex]\[
\boxed{x - 5}
\][/tex]
The quotient is [tex]\( x - 5 \)[/tex], the remainder is [tex]\( 0 \)[/tex], and we express the quotient in standard form as [tex]\( x - 5 \)[/tex].
