Sure, let's solve the division of [tex]\( x^2 + x - 17 \)[/tex] by [tex]\( x - 4 \)[/tex] using polynomial long division step-by-step.
1. Set up the division:
[tex]\[
\frac{x^2 + x - 17}{x - 4}
\][/tex]
2. Divide the leading terms:
- Divide the leading term of the dividend [tex]\( x^2 \)[/tex] by the leading term of the divisor [tex]\( x \)[/tex]:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
- So, the first term of the quotient is [tex]\( x \)[/tex].
3. Multiply and subtract:
- Multiply [tex]\( x \)[/tex] (the quotient term) by the entire divisor [tex]\( x - 4 \)[/tex]:
[tex]\[
x \cdot (x - 4) = x^2 - 4x
\][/tex]
- Subtract this from the original dividend [tex]\( x^2 + x - 17 \)[/tex]:
[tex]\[
(x^2 + x - 17) - (x^2 - 4x) = (x^2 + x - 17) - x^2 + 4x = x + 4x - 17 = 5x - 17
\][/tex]
4. Repeat with the new dividend:
- Now, the new dividend is [tex]\( 5x - 17 \)[/tex].
- Divide the leading term [tex]\( 5x \)[/tex] by the leading term of the divisor [tex]\( x \)[/tex]:
[tex]\[
\frac{5x}{x} = 5
\][/tex]
- So, the next term of the quotient is [tex]\( 5 \)[/tex].
5. Multiply and subtract again:
- Multiply [tex]\( 5 \)[/tex] (the quotient term) by the entire divisor [tex]\( x - 4 \)[/tex]:
[tex]\[
5 \cdot (x - 4) = 5x - 20
\][/tex]
- Subtract this from the current dividend [tex]\( 5x - 17 \)[/tex]:
[tex]\[
(5x - 17) - (5x - 20) = (5x - 17) - 5x + 20 = 5x - 5x + 20 - 17 = 3
\][/tex]
6. Combine the results:
- The quotient is the sum of the terms obtained in steps 2 and 4:
[tex]\[
x + 5
\][/tex]
- The remainder is the result from the subtraction in step 5:
[tex]\[
3
\][/tex]
So, the result of the polynomial long division is:
[tex]\[
\left(x^2 + x - 17\right) \div (x - 4) = x + 5 \text{ with a remainder of 3}
\][/tex]
Written in fraction form, we have:
[tex]\[
\frac{x^2 + x - 17}{x - 4} = x + 5 + \frac{3}{x - 4}
\][/tex]
Therefore, the quotient is [tex]\( x + 5 \)[/tex], and the remainder is 3.
