Answer :
To divide the polynomial [tex]\( 14x^4 + 23x^3 - 221x^2 - 368x - 48 \)[/tex] by [tex]\( 7x + 1 \)[/tex], we'll perform polynomial long division.
### Steps for Polynomial Long Division:
1. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{14x^4}{7x} = 2x^3
\][/tex]
This is the first term of our quotient.
2. Multiply the entire divisor by this first term and subtract from the dividend:
Multiply:
[tex]\[
(7x + 1)(2x^3) = 14x^4 + 2x^3
\][/tex]
Subtract:
[tex]\[
(14x^4 + 23x^3 - 221x^2 - 368x - 48) - (14x^4 + 2x^3) = 21x^3 - 221x^2 - 368x - 48
\][/tex]
3. Repeat the process with the new polynomial:
Divide the new leading term by the leading term of the divisor:
[tex]\[
\frac{21x^3}{7x} = 3x^2
\][/tex]
Multiply and subtract:
[tex]\[
(7x + 1)(3x^2) = 21x^3 + 3x^2
\][/tex]
[tex]\[
(21x^3 - 221x^2 - 368x - 48) - (21x^3 + 3x^2) = -224x^2 - 368x - 48
\][/tex]
4. Continue dividing:
[tex]\[
\frac{-224x^2}{7x} = -32x
\][/tex]
Multiply and subtract:
[tex]\[
(7x + 1)(-32x) = -224x^2 - 32x
\][/tex]
[tex]\[
(-224x^2 - 368x - 48) - (-224x^2 - 32x) = -336x - 48
\][/tex]
5. Last division step:
[tex]\[
\frac{-336x}{7x} = -48
\][/tex]
Multiply and subtract:
[tex]\[
(7x + 1)(-48) = -336x - 48
\][/tex]
[tex]\[
(-336x - 48) - (-336x - 48) = 0
\][/tex]
Since the remainder is 0, the polynomial division ends here.
### Final Answer:
The quotient of the division is [tex]\( 2x^3 + 3x^2 - 32x - 48 \)[/tex] with a remainder of 0.
### Steps for Polynomial Long Division:
1. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{14x^4}{7x} = 2x^3
\][/tex]
This is the first term of our quotient.
2. Multiply the entire divisor by this first term and subtract from the dividend:
Multiply:
[tex]\[
(7x + 1)(2x^3) = 14x^4 + 2x^3
\][/tex]
Subtract:
[tex]\[
(14x^4 + 23x^3 - 221x^2 - 368x - 48) - (14x^4 + 2x^3) = 21x^3 - 221x^2 - 368x - 48
\][/tex]
3. Repeat the process with the new polynomial:
Divide the new leading term by the leading term of the divisor:
[tex]\[
\frac{21x^3}{7x} = 3x^2
\][/tex]
Multiply and subtract:
[tex]\[
(7x + 1)(3x^2) = 21x^3 + 3x^2
\][/tex]
[tex]\[
(21x^3 - 221x^2 - 368x - 48) - (21x^3 + 3x^2) = -224x^2 - 368x - 48
\][/tex]
4. Continue dividing:
[tex]\[
\frac{-224x^2}{7x} = -32x
\][/tex]
Multiply and subtract:
[tex]\[
(7x + 1)(-32x) = -224x^2 - 32x
\][/tex]
[tex]\[
(-224x^2 - 368x - 48) - (-224x^2 - 32x) = -336x - 48
\][/tex]
5. Last division step:
[tex]\[
\frac{-336x}{7x} = -48
\][/tex]
Multiply and subtract:
[tex]\[
(7x + 1)(-48) = -336x - 48
\][/tex]
[tex]\[
(-336x - 48) - (-336x - 48) = 0
\][/tex]
Since the remainder is 0, the polynomial division ends here.
### Final Answer:
The quotient of the division is [tex]\( 2x^3 + 3x^2 - 32x - 48 \)[/tex] with a remainder of 0.
