Answer :
To divide the polynomial [tex]\(4x^4 - 19x^3 + 13x^2 - 7x + 12\)[/tex] by [tex]\(x - 4\)[/tex], we will use polynomial long division. Let's break this down step-by-step.
1. Write the Division Setup:
We set up the division similar to numerical long division:
[tex]\[
\begin{array}{c|c}
x - 4 & 4x^4 - 19x^3 + 13x^2 - 7x + 12 \\ \hline
\end{array}
\][/tex]
2. Divide the First Terms:
Divide the leading term of the dividend [tex]\(4x^4\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], which gives [tex]\(4x^3\)[/tex].
3. Multiply and Subtract:
Multiply [tex]\(4x^3\)[/tex] by the entire divisor [tex]\(x - 4\)[/tex] to get [tex]\(4x^4 - 16x^3\)[/tex].
Subtract this result from the dividend:
[tex]\[
(4x^4 - 19x^3 + 13x^2 - 7x + 12) - (4x^4 - 16x^3) = (-3x^3 + 13x^2 - 7x + 12)
\][/tex]
4. Repeat the Process:
- Divide [tex]\(-3x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-3x^2\)[/tex].
- Multiply [tex]\(-3x^2\)[/tex] by [tex]\(x - 4\)[/tex] to get [tex]\(-3x^3 + 12x^2\)[/tex].
- Subtract:
[tex]\[
(-3x^3 + 13x^2 - 7x + 12) - (-3x^3 + 12x^2) = x^2 - 7x + 12
\][/tex]
5. Continue Dividing:
- Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(x\)[/tex].
- Multiply [tex]\(x\)[/tex] by [tex]\(x - 4\)[/tex] to get [tex]\(x^2 - 4x\)[/tex].
- Subtract:
[tex]\[
(x^2 - 7x + 12) - (x^2 - 4x) = -3x + 12
\][/tex]
6. Final Division:
- Divide [tex]\(-3x\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-3\)[/tex].
- Multiply [tex]\(-3\)[/tex] by [tex]\(x - 4\)[/tex] to get [tex]\(-3x + 12\)[/tex].
- Subtract:
[tex]\[
(-3x + 12) - (-3x + 12) = 0
\][/tex]
7. Conclusion:
The quotient is [tex]\(4x^3 - 3x^2 + x - 3\)[/tex] and the remainder is [tex]\(0\)[/tex]. Therefore, the division results in:
[tex]\[
\frac{4x^4 - 19x^3 + 13x^2 - 7x + 12}{x - 4} = 4x^3 - 3x^2 + x - 3
\][/tex]
This means the polynomial [tex]\(4x^4 - 19x^3 + 13x^2 - 7x + 12\)[/tex] divides evenly by [tex]\(x - 4\)[/tex] with no remainder.
1. Write the Division Setup:
We set up the division similar to numerical long division:
[tex]\[
\begin{array}{c|c}
x - 4 & 4x^4 - 19x^3 + 13x^2 - 7x + 12 \\ \hline
\end{array}
\][/tex]
2. Divide the First Terms:
Divide the leading term of the dividend [tex]\(4x^4\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex], which gives [tex]\(4x^3\)[/tex].
3. Multiply and Subtract:
Multiply [tex]\(4x^3\)[/tex] by the entire divisor [tex]\(x - 4\)[/tex] to get [tex]\(4x^4 - 16x^3\)[/tex].
Subtract this result from the dividend:
[tex]\[
(4x^4 - 19x^3 + 13x^2 - 7x + 12) - (4x^4 - 16x^3) = (-3x^3 + 13x^2 - 7x + 12)
\][/tex]
4. Repeat the Process:
- Divide [tex]\(-3x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-3x^2\)[/tex].
- Multiply [tex]\(-3x^2\)[/tex] by [tex]\(x - 4\)[/tex] to get [tex]\(-3x^3 + 12x^2\)[/tex].
- Subtract:
[tex]\[
(-3x^3 + 13x^2 - 7x + 12) - (-3x^3 + 12x^2) = x^2 - 7x + 12
\][/tex]
5. Continue Dividing:
- Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(x\)[/tex].
- Multiply [tex]\(x\)[/tex] by [tex]\(x - 4\)[/tex] to get [tex]\(x^2 - 4x\)[/tex].
- Subtract:
[tex]\[
(x^2 - 7x + 12) - (x^2 - 4x) = -3x + 12
\][/tex]
6. Final Division:
- Divide [tex]\(-3x\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-3\)[/tex].
- Multiply [tex]\(-3\)[/tex] by [tex]\(x - 4\)[/tex] to get [tex]\(-3x + 12\)[/tex].
- Subtract:
[tex]\[
(-3x + 12) - (-3x + 12) = 0
\][/tex]
7. Conclusion:
The quotient is [tex]\(4x^3 - 3x^2 + x - 3\)[/tex] and the remainder is [tex]\(0\)[/tex]. Therefore, the division results in:
[tex]\[
\frac{4x^4 - 19x^3 + 13x^2 - 7x + 12}{x - 4} = 4x^3 - 3x^2 + x - 3
\][/tex]
This means the polynomial [tex]\(4x^4 - 19x^3 + 13x^2 - 7x + 12\)[/tex] divides evenly by [tex]\(x - 4\)[/tex] with no remainder.
