Answer :
To divide the polynomial [tex]\(\frac{15x^6 - 9x^5 + 56x^4 + 70x^3 + 73x - 23}{3x^3 + 7x + 11}\)[/tex], we will use polynomial long division. Here’s a detailed step-by-step solution:
1. Setup for Division:
The polynomial division is similar to long division of numbers. We divide the leading term of the numerator by the leading term of the denominator.
2. First Division Step:
Divide [tex]\( 15x^6 \)[/tex] by [tex]\( 3x^3 \)[/tex]:
[tex]\[
\frac{15x^6}{3x^3} = 5x^3
\][/tex]
Multiply [tex]\( 5x^3 \)[/tex] by the entire denominator [tex]\( 3x^3 + 7x + 11 \)[/tex]:
[tex]\[
5x^3 \cdot (3x^3 + 7x + 11) = 15x^6 + 35x^4 + 55x^3
\][/tex]
Subtract this product from the original polynomial:
[tex]\[
(15x^6 - 9x^5 + 56x^4 + 70x^3 + 73x - 23) - (15x^6 + 35x^4 + 55x^3) = -9x^5 + 21x^4 + 15x^3 + 73x - 23
\][/tex]
3. Second Division Step:
Divide [tex]\( -9x^5 \)[/tex] by [tex]\( 3x^3 \)[/tex]:
[tex]\[
\frac{-9x^5}{3x^3} = -3x^2
\][/tex]
Multiply [tex]\( -3x^2 \)[/tex] by the entire denominator [tex]\( 3x^3 + 7x + 11 \)[/tex]:
[tex]\[
-3x^2 \cdot (3x^3 + 7x + 11) = -9x^5 - 21x^3 - 33x^2
\][/tex]
Subtract this product from the current polynomial:
[tex]\[
(-9x^5 + 21x^4 + 15x^3 + 73x - 23) - (-9x^5 - 21x^3 - 33x^2) = 21x^4 + 36x^3 + 33x^2 + 73x - 23
\][/tex]
4. Third Division Step:
Divide [tex]\( 21x^4 \)[/tex] by [tex]\( 3x^3 \)[/tex]:
[tex]\[
\frac{21x^4}{3x^3} = 7x
\][/tex]
Multiply [tex]\( 7x \)[/tex] by the entire denominator [tex]\( 3x^3 + 7x + 11 \)[/tex]:
[tex]\[
7x \cdot (3x^3 + 7x + 11) = 21x^4 + 49x^2 + 77x
\][/tex]
Subtract this product from the current polynomial:
[tex]\[
(21x^4 + 36x^3 + 33x^2 + 73x - 23) - (21x^4 + 49x^2 + 77x) = 36x^3 - 16x^2 - 4x - 23
\][/tex]
5. Fourth Division Step:
Divide [tex]\( 36x^3 \)[/tex] by [tex]\( 3x^3 \)[/tex]:
[tex]\[
\frac{36x^3}{3x^3} = 12
\][/tex]
Multiply [tex]\( 12 \)[/tex] by the entire denominator [tex]\( 3x^3 + 7x + 11 \)[/tex]:
[tex]\[
12 \cdot (3x^3 + 7x + 11) = 36x^3 + 84x + 132
\][/tex]
Subtract this product from the current polynomial:
[tex]\[
(36x^3 - 16x^2 - 4x - 23) - (36x^3 + 84x + 132) = -16x^2 - 88x - 155
\][/tex]
6. Final Result:
The quotient is [tex]\( 5x^3 - 3x^2 + 7x + 12 \)[/tex] and the remainder is [tex]\( -16x^2 - 88x - 155 \)[/tex].
Therefore, the division of [tex]\(\frac{15x^6 - 9x^5 + 56x^4 + 70x^3 + 73x - 23}{3x^3 + 7x + 11}\)[/tex] gives:
[tex]\[
5x^3 - 3x^2 + 7x + 12 + \frac{-16x^2 - 88x - 155}{3x^3 + 7x + 11}
\][/tex]
1. Setup for Division:
The polynomial division is similar to long division of numbers. We divide the leading term of the numerator by the leading term of the denominator.
2. First Division Step:
Divide [tex]\( 15x^6 \)[/tex] by [tex]\( 3x^3 \)[/tex]:
[tex]\[
\frac{15x^6}{3x^3} = 5x^3
\][/tex]
Multiply [tex]\( 5x^3 \)[/tex] by the entire denominator [tex]\( 3x^3 + 7x + 11 \)[/tex]:
[tex]\[
5x^3 \cdot (3x^3 + 7x + 11) = 15x^6 + 35x^4 + 55x^3
\][/tex]
Subtract this product from the original polynomial:
[tex]\[
(15x^6 - 9x^5 + 56x^4 + 70x^3 + 73x - 23) - (15x^6 + 35x^4 + 55x^3) = -9x^5 + 21x^4 + 15x^3 + 73x - 23
\][/tex]
3. Second Division Step:
Divide [tex]\( -9x^5 \)[/tex] by [tex]\( 3x^3 \)[/tex]:
[tex]\[
\frac{-9x^5}{3x^3} = -3x^2
\][/tex]
Multiply [tex]\( -3x^2 \)[/tex] by the entire denominator [tex]\( 3x^3 + 7x + 11 \)[/tex]:
[tex]\[
-3x^2 \cdot (3x^3 + 7x + 11) = -9x^5 - 21x^3 - 33x^2
\][/tex]
Subtract this product from the current polynomial:
[tex]\[
(-9x^5 + 21x^4 + 15x^3 + 73x - 23) - (-9x^5 - 21x^3 - 33x^2) = 21x^4 + 36x^3 + 33x^2 + 73x - 23
\][/tex]
4. Third Division Step:
Divide [tex]\( 21x^4 \)[/tex] by [tex]\( 3x^3 \)[/tex]:
[tex]\[
\frac{21x^4}{3x^3} = 7x
\][/tex]
Multiply [tex]\( 7x \)[/tex] by the entire denominator [tex]\( 3x^3 + 7x + 11 \)[/tex]:
[tex]\[
7x \cdot (3x^3 + 7x + 11) = 21x^4 + 49x^2 + 77x
\][/tex]
Subtract this product from the current polynomial:
[tex]\[
(21x^4 + 36x^3 + 33x^2 + 73x - 23) - (21x^4 + 49x^2 + 77x) = 36x^3 - 16x^2 - 4x - 23
\][/tex]
5. Fourth Division Step:
Divide [tex]\( 36x^3 \)[/tex] by [tex]\( 3x^3 \)[/tex]:
[tex]\[
\frac{36x^3}{3x^3} = 12
\][/tex]
Multiply [tex]\( 12 \)[/tex] by the entire denominator [tex]\( 3x^3 + 7x + 11 \)[/tex]:
[tex]\[
12 \cdot (3x^3 + 7x + 11) = 36x^3 + 84x + 132
\][/tex]
Subtract this product from the current polynomial:
[tex]\[
(36x^3 - 16x^2 - 4x - 23) - (36x^3 + 84x + 132) = -16x^2 - 88x - 155
\][/tex]
6. Final Result:
The quotient is [tex]\( 5x^3 - 3x^2 + 7x + 12 \)[/tex] and the remainder is [tex]\( -16x^2 - 88x - 155 \)[/tex].
Therefore, the division of [tex]\(\frac{15x^6 - 9x^5 + 56x^4 + 70x^3 + 73x - 23}{3x^3 + 7x + 11}\)[/tex] gives:
[tex]\[
5x^3 - 3x^2 + 7x + 12 + \frac{-16x^2 - 88x - 155}{3x^3 + 7x + 11}
\][/tex]
