To divide the polynomial [tex]\( f(x) = x^2 - 7x - 18 \)[/tex] by [tex]\( d(x) = x - 4 \)[/tex], we use polynomial division. Let's perform the division step-by-step:
1. Setup the long division: Arrange the polynomials in standard form.
[tex]\[
\frac{x^2 - 7x - 18}{x - 4}
\][/tex]
2. Divide the leading term of the numerator by the leading term of the denominator.
The leading term of the numerator is [tex]\( x^2 \)[/tex] and the leading term of the denominator is [tex]\( x \)[/tex].
[tex]\[
\frac{x^2}{x} = x
\][/tex]
So, the first term of the quotient [tex]\( Q(x) \)[/tex] is [tex]\( x \)[/tex].
3. Multiply this term by the divisor [tex]\( d(x) \)[/tex] and subtract the result from the original polynomial.
[tex]\[
(x - 4) \cdot x = x^2 - 4x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
x^2 - 7x - 18 - (x^2 - 4x) = (x^2 - 7x - 18) - (x^2 - 4x) = - 3x - 18
\][/tex]
4. Repeat the process with the new polynomial. Divide the leading term of the new polynomial by the leading term of the divisor.
The leading term of the new polynomial is [tex]\( -3x \)[/tex], so
[tex]\[
\frac{-3x}{x} = -3
\][/tex]
Now, the second term of the quotient is [tex]\( -3 \)[/tex].
5. Multiply this term by the divisor and subtract the result from the new polynomial.
[tex]\[
(x - 4) \cdot -3 = -3x + 12
\][/tex]
Subtract this from the new polynomial:
[tex]\[
-3x - 18 - (-3x + 12) = (-3x - 18) - (-3x + 12) = -18 - 12 = -30
\][/tex]
At this point, the remainder is found to be a constant, which means the division process is complete. The quotient [tex]\( Q(x) \)[/tex] is [tex]\( x - 3 \)[/tex] and the remainder [tex]\( R(x) \)[/tex] is [tex]\( -30 \)[/tex].
So, the division of [tex]\( f(x) \)[/tex] by [tex]\( d(x) \)[/tex] can be expressed as:
[tex]\[
\begin{array}{c}
\frac{f(x)}{d(x)} = Q(x) + \frac{R(x)}{d(x)} \\
\frac{f(x)}{d(x)} = \frac{x^2 - 7x - 18}{x - 4} = x - 3 + \frac{-30}{x - 4} \\
R(x) = -30
\end{array}
\][/tex]
Thus, our result is:
[tex]\[
\frac{x^2 - 7x - 18}{x - 4} = x - 3 + \frac{-30}{x - 4}
\][/tex]
and
[tex]\[
R(x) = -30
\][/tex]
