Answer :
To find the quotient and remainder of [tex]\( \left(x^5 - x^3 + x - 5\right) \div (x - 2) \)[/tex], we can perform polynomial long division.
Let's go through the steps:
1. Divide the first term: Divide the first term of the dividend, [tex]\( x^5 \)[/tex], by the first term of the divisor, [tex]\( x \)[/tex]. This gives [tex]\( x^4 \)[/tex].
2. Multiply and subtract: Multiply [tex]\( x^4 \)[/tex] by the entire divisor, [tex]\( x - 2 \)[/tex], to get [tex]\( x^5 - 2x^4 \)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^5 - x^3 + x - 5) - (x^5 - 2x^4) = 2x^4 - x^3 + x - 5
\][/tex]
3. Repeat: Divide the new leading term, [tex]\( 2x^4 \)[/tex], by the leading term of the divisor, [tex]\( x \)[/tex], which gives [tex]\( 2x^3 \)[/tex].
4. Multiply and subtract: Multiply [tex]\( 2x^3 \)[/tex] by [tex]\( x - 2 \)[/tex] to get [tex]\( 2x^4 - 4x^3 \)[/tex]. Subtract:
[tex]\[
(2x^4 - x^3 + x - 5) - (2x^4 - 4x^3) = 3x^3 + x - 5
\][/tex]
5. Continue the process: Divide [tex]\( 3x^3 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( 3x^2 \)[/tex].
6. Multiply and subtract: Multiply [tex]\( 3x^2 \)[/tex] by [tex]\( x - 2 \)[/tex] to get [tex]\( 3x^3 - 6x^2 \)[/tex]. Subtract:
[tex]\[
(3x^3 + x - 5) - (3x^3 - 6x^2) = 6x^2 + x - 5
\][/tex]
7. Continue dividing: Divide [tex]\( 6x^2 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( 6x \)[/tex].
8. Multiply and subtract: Multiply [tex]\( 6x \)[/tex] by [tex]\( x - 2 \)[/tex] to get [tex]\( 6x^2 - 12x \)[/tex]. Subtract:
[tex]\[
(6x^2 + x - 5) - (6x^2 - 12x) = 13x - 5
\][/tex]
9. Final step: Divide [tex]\( 13x \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( 13 \)[/tex].
10. Multiply and subtract: Multiply [tex]\( 13 \)[/tex] by [tex]\( x - 2 \)[/tex] to get [tex]\( 13x - 26 \)[/tex]. Subtract:
[tex]\[
(13x - 5) - (13x - 26) = 21
\][/tex]
The polynomial division is complete, and the quotient is [tex]\( x^4 + 2x^3 + 3x^2 + 6x + 13 \)[/tex], and the remainder is [tex]\( 21 \)[/tex].
Therefore, the answer is:
d. [tex]\( x^4 + 2x^3 + 3x^2 + 6x + 13 ; 21 \)[/tex]
Let's go through the steps:
1. Divide the first term: Divide the first term of the dividend, [tex]\( x^5 \)[/tex], by the first term of the divisor, [tex]\( x \)[/tex]. This gives [tex]\( x^4 \)[/tex].
2. Multiply and subtract: Multiply [tex]\( x^4 \)[/tex] by the entire divisor, [tex]\( x - 2 \)[/tex], to get [tex]\( x^5 - 2x^4 \)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^5 - x^3 + x - 5) - (x^5 - 2x^4) = 2x^4 - x^3 + x - 5
\][/tex]
3. Repeat: Divide the new leading term, [tex]\( 2x^4 \)[/tex], by the leading term of the divisor, [tex]\( x \)[/tex], which gives [tex]\( 2x^3 \)[/tex].
4. Multiply and subtract: Multiply [tex]\( 2x^3 \)[/tex] by [tex]\( x - 2 \)[/tex] to get [tex]\( 2x^4 - 4x^3 \)[/tex]. Subtract:
[tex]\[
(2x^4 - x^3 + x - 5) - (2x^4 - 4x^3) = 3x^3 + x - 5
\][/tex]
5. Continue the process: Divide [tex]\( 3x^3 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( 3x^2 \)[/tex].
6. Multiply and subtract: Multiply [tex]\( 3x^2 \)[/tex] by [tex]\( x - 2 \)[/tex] to get [tex]\( 3x^3 - 6x^2 \)[/tex]. Subtract:
[tex]\[
(3x^3 + x - 5) - (3x^3 - 6x^2) = 6x^2 + x - 5
\][/tex]
7. Continue dividing: Divide [tex]\( 6x^2 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( 6x \)[/tex].
8. Multiply and subtract: Multiply [tex]\( 6x \)[/tex] by [tex]\( x - 2 \)[/tex] to get [tex]\( 6x^2 - 12x \)[/tex]. Subtract:
[tex]\[
(6x^2 + x - 5) - (6x^2 - 12x) = 13x - 5
\][/tex]
9. Final step: Divide [tex]\( 13x \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( 13 \)[/tex].
10. Multiply and subtract: Multiply [tex]\( 13 \)[/tex] by [tex]\( x - 2 \)[/tex] to get [tex]\( 13x - 26 \)[/tex]. Subtract:
[tex]\[
(13x - 5) - (13x - 26) = 21
\][/tex]
The polynomial division is complete, and the quotient is [tex]\( x^4 + 2x^3 + 3x^2 + 6x + 13 \)[/tex], and the remainder is [tex]\( 21 \)[/tex].
Therefore, the answer is:
d. [tex]\( x^4 + 2x^3 + 3x^2 + 6x + 13 ; 21 \)[/tex]
