Question

use long division to divide the following polynomials.
$(8n^2 + 50n + 60) \div (4n + 7)$
show your work here
enter your answer

use synthetic division to divide the following expressions. then enter your solution as a polynomial.
$(7k^3 - 13k^2 + 11k - 10) \div (k - 1)$
show your work here
hint: to add an exponent $(x^y)$, type \exponent\ or press \^\

1 | 7 -13 11 -10
\downarrow \square \square \square

Explanation:

First Sub - Question: Polynomial Long Division \((8n^{2}+50n + 60)\div(4n + 7)\)

Step 1: Divide the leading terms

Divide the leading term of the dividend \(8n^{2}\) by the leading term of the divisor \(4n\). So, \(\frac{8n^{2}}{4n}=2n\). This is the first term of the quotient.

Step 2: Multiply and subtract

Multiply the divisor \((4n + 7)\) by \(2n\): \(2n\times(4n + 7)=8n^{2}+14n\).
Subtract this from the dividend: \((8n^{2}+50n + 60)-(8n^{2}+14n)=36n + 60\).

Step 3: Divide the new leading terms

Divide the leading term of \(36n\) (from the new dividend) by the leading term of the divisor \(4n\). So, \(\frac{36n}{4n}=9\). This is the second term of the quotient.

Step 4: Multiply and subtract again

Multiply the divisor \((4n + 7)\) by \(9\): \(9\times(4n + 7)=36n+63\).
Subtract this from \(36n + 60\): \((36n + 60)-(36n + 63)=- 3\).

Step 1: Set up synthetic division

For synthetic division with divisor \(k - 1\), we use the root \(r = 1\). The coefficients of the dividend \(7k^{3}-13k^{2}+11k - 10\) are \(7,-13,11,-10\).

Step 2: Bring down the first coefficient

Bring down the first coefficient \(7\) as it is.

Step 3: Multiply and add

Multiply the number we just brought down (\(7\)) by the root \(1\): \(7\times1 = 7\).
Add this to the next coefficient: \(-13+7=-6\).

Step 4: Multiply and add again

Multiply \(-6\) by \(1\): \(-6\times1=-6\).
Add this to the next coefficient: \(11+(-6) = 5\).

Step 5: Multiply and add a third time

Multiply \(5\) by \(1\): \(5\times1 = 5\).
Add this to the last coefficient: \(-10 + 5=-5\).
The coefficients of the quotient polynomial (from left to right) are \(7,-6,5\) and the remainder is \(-5\). Since the original polynomial is of degree \(3\), the quotient polynomial is of degree \(2\). So the quotient is \(7k^{2}-6k + 5\) and the remainder is \(-5\).

Answer:

The quotient is \(2n + 9\) and the remainder is \(-3\), so \((8n^{2}+50n + 60)\div(4n + 7)=2n + 9-\frac{3}{4n + 7}\)

Second Sub - Question: Polynomial Synthetic Division \((7k^{3}-13k^{2}+11k - 10)\div(k - 1)\)