Question

What is the answer to (m^2-7m-11)/(m-8)

Answer 1

Answer:

The final quotient is $\bold{m+1-\frac{3}{m-8}}$.

Step-by-step explanation:

This problem can be solved using two methods: long polynomial division and synthetic division.

Method 1: Long Polynomial Division

We can use long polynomial division to solve the equation. The steps are detailed below the work.

$\displaystyle{\begin{array}{r}\bold{m+1-\frac{3}{m-8}\phantom{)}\\m-8{\overline{\smash{\big)}\,m^2-7m-11\phantom{)}}}\\\underline{-~\phantom{(}m^2-8m\phantom{-b)}}\\m-11\phantom{)}\\\underline{-~\phantom{()}m-8}\\-3\phantom{)}\end{array}}$

Steps

1. Set up your division. Place the divisor (the binomial) on the left hand side of the division symbol and the trinomial (or polynomial depending on the amount of terms) on the right hand side of the equation.
2. Divide the divisor by the first term of the trinomial: $\frac{m^2}{m}=m$
3. Place the result above the first term of the trinomial (above the bar and directly above the first term of the polynomial.
4. Multiply the m by the divisor: $m(m-8) = m^2-8m$.
5. Subtract this from the original trinomial:
6. $(m^2-7m-11)-(m^2-8m)=m - 11$.
7. Set this below the equation and then divide the leading term of the divisor by the leading term of the remainder: $\frac{m}{m}=1$.
8. Place this to the right of the m located above - in this case, we add because our result is positive.
9. Multiply that by the divisor: $1(m-8)=m-8$.
10. Set this below the remainder from the previous division and subtract: $(m-11)-(m-8)=-3$.
11. Finally, divide the original divisor by the remainder: $\frac{-3}{m-8}$.
12. This divisor needs simplified: $-\frac{3}{m-8}$.
13. Our final new polynomial (once evaluated), is $\bold{m+1-\frac{3}{m-8}}$.

Method 2: Synthetic Division

We can also use synthetic division to solve the polynomial.

Table 1 (actual)        Table 2 (model)

$\begin{array}{c|ccc}8 & 1 & -7 & -11 \\ & \downarrow & 8 & 8 \\ \cline{2-4} \multicolumn{1}{c}{} & 1 & 1 & -3\end{array}$      $\begin{array}{c|ccc}a & b & c & d \\ & \downarrow & e & f \\ \cline{2-4} \multicolumn{1}{c}{} & g & h & i\end{array}$

Steps

1. Determine the coefficients of the polynomial: $m^2-7m-11$. These are 1, -7, and -11.
2. We need to determine the zero. We do this by setting the divisor equal to zero: $m-8=0\rightarrow m=8$. Place this value in the upper left hand corner (in place of a).
3. For the first term, we just drop the coefficient below the bar. Therefore, g becomes 1.
4. Multiply a by g to get e : $8 \times 1 = 8$. Therefore, e becomes 8.
5. Add c to e : $-7 + 8 = 1$. Therefore, h becomes 1.
6. Multiply a by h to get f : $8 \times 1 = 8$. Therefore, f becomes 8.
7. Add d to f to get i : $-11 + 8 = -3$. Therefore, i becomes -3.
8. Our new coefficients are 1, 1, and -3. We lose one power (if the first term is squared, it now does not have an exponent).
9. Use the new coefficients to get our new equation: $m+1-3$.
10. Finally, the last term is the remainder. So, we divide this by the divisor: $m+1-\frac{3}{m-8}$.
11. Our final expression is $\bold{m+1-\frac{3}{m-8}}$.