5. $(3x^2 - 3x + 5) div (x - 6)$ 7. $(4x^3 - 6x^2 - 4x + 8) div (2x - 1)$ 6. $(2x^3 + 6x^2 - 6x + 2) div (2x - 3)$ 8. $(2x^3 - 5x^2 + 3x - 6) div (x + 4)$

Question

5.  $(3x^2 - 3x + 5) div (x - 6)$  7.  $(4x^3 - 6x^2 - 4x + 8) div (2x - 1)$  6.  $(2x^3 + 6x^2 - 6x + 2) div (2x - 3)$  8.  $(2x^3 - 5x^2 + 3x - 6) div (x + 4)$

Accepted Answer

### Problem 5: $(3x^2 - 3x + 5) \div (x - 6)$
#### Step 1: Use polynomial long division
Divide the first term of the dividend $3x^2$ by the first term of the divisor $x$ to get $3x$. Multiply the divisor $(x - 6)$ by $3x$ to get $3x^2 - 18x$. Subtract this from the dividend:
$$
\begin{align*}
(3x^2 - 3x + 5) - (3x^2 - 18x) &= 3x^2 - 3x + 5 - 3x^2 + 18x\
&= 15x + 5
\end{align*}
$$
#### Step 2: Divide the new first term $15x$ by $x$ to get $15$. Multiply the divisor $(x - 6)$ by $15$ to get $15x - 90$. Subtract this from $15x + 5$:
$$
(15x + 5) - (15x - 90) = 15x + 5 - 15x + 90 = 95
$$
So, $(3x^2 - 3x + 5) \div (x - 6) = 3x + 15 + \frac{95}{x - 6}$

### Problem 6: $(2x^3 + 6x^2 - 6x + 2) \div (2x - 3)$
#### Step 1: Divide the first term of the dividend $2x^3$ by the first term of the divisor $2x$ to get $x^2$. Multiply the divisor $(2x - 3)$ by $x^2$ to get $2x^3 - 3x^2$. Subtract this from the dividend:
$$
\begin{align*}
(2x^3 + 6x^2 - 6x + 2) - (2x^3 - 3x^2) &= 2x^3 + 6x^2 - 6x + 2 - 2x^3 + 3x^2\
&= 9x^2 - 6x + 2
\end{align*}
$$
#### Step 2: Divide the new first term $9x^2$ by $2x$ to get $\frac{9}{2}x$. Multiply the divisor $(2x - 3)$ by $\frac{9}{2}x$ to get $9x^2 - \frac{27}{2}x$. Subtract this from $9x^2 - 6x + 2$:
$$
\begin{align*}
(9x^2 - 6x + 2) - (9x^2 - \frac{27}{2}x) &= 9x^2 - 6x + 2 - 9x^2 + \frac{27}{2}x\
&= \frac{15}{2}x + 2
\end{align*}
$$
#### Step 3: Divide the new first term $\frac{15}{2}x$ by $2x$ to get $\frac{15}{4}$. Multiply the divisor $(2x - 3)$ by $\frac{15}{4}$ to get $\frac{15}{2}x - \frac{45}{4}$. Subtract this from $\frac{15}{2}x + 2$:
$$
\begin{align*}
(\frac{15}{2}x + 2) - (\frac{15}{2}x - \frac{45}{4}) &= \frac{15}{2}x + 2 - \frac{15}{2}x + \frac{45}{4}\
&= \frac{8}{4} + \frac{45}{4}\
&= \frac{53}{4}
\end{align*}
$$
So, $(2x^3 + 6x^2 - 6x + 2) \div (2x - 3) = x^2 + \frac{9}{2}x + \frac{15}{4} + \frac{53}{4(2x - 3)}$

### Problem 7: $(4x^3 - 6x^2 - 4x + 8) \div (2x - 1)$
#### Step 1: Divide the first term of the dividend $4x^3$ by the first term of the divisor $2x$ to get $2x^2$. Multiply the divisor $(2x - 1)$ by $2x^2$ to get $4x^3 - 2x^2$. Subtract this from the dividend:
$$
\begin{align*}
(4x^3 - 6x^2 - 4x + 8) - (4x^3 - 2x^2) &= 4x^3 - 6x^2 - 4x + 8 - 4x^3 + 2x^2\
&= -4x^2 - 4x + 8
\end{align*}
$$
#### Step 2: Divide the new first term $-4x^2$ by $2x$ to get $-2x$. Multiply the divisor $(2x - 1)$ by $-2x$ to get $-4x^2 + 2x$. Subtract this from $-4x^2 - 4x + 8$:
$$
\begin{align*}
(-4x^2 - 4x + 8) - (-4x^2 + 2x) &= -4x^2 - 4x + 8 + 4x^2 - 2x\
&= -6x + 8
\end{align*}
$$
#### Step 3: Divide the new first term $-6x$ by $2x$ to get $-3$. Multiply the divisor $(2x - 1)$ by $-3$ to get $-6x + 3$. Subtract this from $-6x + 8$:
$$
(-6x + 8) - (-6x + 3) = -6x + 8 + 6x - 3 = 5
$$
So, $(4x^3 - 6x^2 - 4x + 8) \div (2x - 1) = 2x^2 - 2x - 3 + \frac{5}{2x - 1}$

### Problem 8: $(2x^3 - 5x^2 + 3x - 6) \div (x + 4)$
#### Step 1: Divide the first term of the dividend $2x^3$ by the first term of the divisor $x$ to get $2x^2$. Multiply the divisor $(x + 4)$ by $2x^2$ to get $2x^3 + 8x^2$. Subtract this from the dividend:
$$
\begin{align*}
(2x^3 - 5x^2 + 3x - 6) - (2x^3 + 8x^2) &= 2x^3 - 5x^2 + 3x - 6 - 2x^3 - 8x^2\
&= -13x^2 + 3x - 6
\end{align*}
$$
#### Step 2: Divide the new first term $-13x^2$ by $x$ to get $-13x$. Multiply the divisor $(x + 4)$ by $-13x$ to get $-13x^2 - 52x$. Subtract this from $-13x^2 + 3x - 6$:
$$
\begin{align*}
(-13x^2 + 3x - 6) - (-13x^2 - 52x) &= -13x^2 + 3x - 6 + 13x^2 + 52x\
&= 55x - 6
\end{align*}
$$
#### Step 3: Divide the new first term $55x$ by $x$ to get $55$. Multiply the divisor $(x + 4)$ by $55$ to get $55x + 220$. Subtract this from $55x - 6$:
$$
(55x - 6) - (55x + 220) = 55x - 6 - 55x - 220 = -226
$$
So, $(2x^3 - 5x^2 + 3x - 6) \div (x + 4) = 2x^2 - 13x + 55 - \frac{226}{x + 4}$

### Final Answers:
5. $3x + 15 + \frac{95}{x - 6}$
6. $x^2 + \frac{9}{2}x + \frac{15}{4} + \frac{53}{4(2x - 3)}$
7. $2x^2 - 2x - 3 + \frac{5}{2x - 1}$
8. $2x^2 - 13x + 55 - \frac{226}{x + 4}$

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QUESTION IMAGE

Question

Explanation:

Problem 5: \((3x^2 - 3x + 5) \div (x - 6)\)

Step 1: Use polynomial long division

Step 2: Divide the new first term \(15x\) by \(x\) to get \(15\). Multiply the divisor \((x - 6)\) by \(15\) to get \(15x - 90\). Subtract this from \(15x + 5\):

Problem 6: \((2x^3 + 6x^2 - 6x + 2) \div (2x - 3)\)

Step 1: Divide the first term of the dividend \(2x^3\) by the first term of the divisor \(2x\) to get \(x^2\). Multiply the divisor \((2x - 3)\) by \(x^2\) to get \(2x^3 - 3x^2\). Subtract this from the dividend:

Step 2: Divide the new first term \(9x^2\) by \(2x\) to get \(\frac{9}{2}x\). Multiply the divisor \((2x - 3)\) by \(\frac{9}{2}x\) to get \(9x^2 - \frac{27}{2}x\). Subtract this from \(9x^2 - 6x + 2\):

Step 3: Divide the new first term \(\frac{15}{2}x\) by \(2x\) to get \(\frac{15}{4}\). Multiply the divisor \((2x - 3)\) by \(\frac{15}{4}\) to get \(\frac{15}{2}x - \frac{45}{4}\). Subtract this from \(\frac{15}{2}x + 2\):

Problem 7: \((4x^3 - 6x^2 - 4x + 8) \div (2x - 1)\)

Step 1: Divide the first term of the dividend \(4x^3\) by the first term of the divisor \(2x\) to get \(2x^2\). Multiply the divisor \((2x - 1)\) by \(2x^2\) to get \(4x^3 - 2x^2\). Subtract this from the dividend:

Step 2: Divide the new first term \(-4x^2\) by \(2x\) to get \(-2x\). Multiply the divisor \((2x - 1)\) by \(-2x\) to get \(-4x^2 + 2x\). Subtract this from \(-4x^2 - 4x + 8\):

Step 3: Divide the new first term \(-6x\) by \(2x\) to get \(-3\). Multiply the divisor \((2x - 1)\) by \(-3\) to get \(-6x + 3\). Subtract this from \(-6x + 8\):

Problem 8: \((2x^3 - 5x^2 + 3x - 6) \div (x + 4)\)

Step 1: Divide the first term of the dividend \(2x^3\) by the first term of the divisor \(x\) to get \(2x^2\). Multiply the divisor \((x + 4)\) by \(2x^2\) to get \(2x^3 + 8x^2\). Subtract this from the dividend:

Step 2: Divide the new first term \(-13x^2\) by \(x\) to get \(-13x\). Multiply the divisor \((x + 4)\) by \(-13x\) to get \(-13x^2 - 52x\). Subtract this from \(-13x^2 + 3x - 6\):

Answer: