Use the long division method to find the result when 8, x, cubed, plus, 2, x, squared, minus, 14, x, plus, 38x

Question

3
+2x
2
−14x+3 is divided by 4, x, plus, 54x+5. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+
b(x)
r(x)
​
.



Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, minus, 20, x, squared, minus, 27, x, plus, 19
4x
3
−20x
2
−27x+19 is divided by
x, minus, 6
x−6. If there is a remainder, express the result in the form
q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction
q(x)+
b(x)
r(x)
​
.
Use Long Division:
Use Long Division:

4, x, squared
4x
2
plus, 4, x
+4x
minus, 3
−3

x
x
minus, 6
−6
right parenthesis
)
4, x, cubed
4x
3
minus, 20, x, squared
−20x
2
minus, 27, x
−27x
plus, 19
+19

start superscript, minus, end superscript, 4, x, cubed
−
4x
3
minus, start superscript, plus, end superscript, 24, x, squared
−
+
24x
2

4, x, squared
4x
2
minus, 27, x
−27x
plus, 19
+19

start superscript, minus, end superscript, 4, x, squared
−
4x
2
minus, start superscript, plus, end superscript, 24, x
−
+
24x

minus, 3, x
−3x
plus, 19
+19

minus, start superscript, plus, end superscript, 3, x
−
+
3x
plus, start superscript, minus, end superscript, 18
+
−
18

1
1

Answer: , 4, x, squared, plus, 4, x, minus, 3, plus, start fraction, 1, divided by, x, minus, 6, end fraction
Answer: 4x
2
+4x−3+
x−6
1
​
Form: Quotient plus , start fraction, Remainder, divided by, Divisor, end fraction
Form: Quotient +
Divisor
Remainder
​

Answer 1

To divide \(8x^3 + 2x^2 - 14x + 38\) by \(4x + 5\) using the long division method, we will proceed as follows:

Step 1: Set up the long division

              _________
4x + 5 |  8x^3 + 2x^2 - 14x + 38

Step 2: Divide the leading terms

Divide the leading term \(8x^3\) by \(4x\): \[ \frac{8x^3}{4x} = 2x^2 \]

Step 3: Multiply and subtract

Multiply \(2x^2\) by \(4x + 5\): \[ 2x^2 \cdot (4x + 5) = 8x^3 + 10x^2 \] Now, subtract this from \(8x^3 + 2x^2\): \[ \begin{align*} (8x^3 + 2x^2) - (8x^3 + 10x^2) &= -8x^2 \end{align*} \] Bring down the next term \(-14x\): \[ -8x^2 - 14x \]

Step 4: Repeat the process

Now divide \(-8x^2\) by \(4x\): \[ \frac{-8x^2}{4x} = -2x \] Multiply \(-2x\) by \(4x + 5\): \[ -2x \cdot (4x + 5) = -8x^2 - 10x \] Now subtract this from \(-8x^2 - 14x\): \[ \begin{align*} (-8x^2 - 14x) - (-8x^2 - 10x) &= -4x \end{align*} \] Bring down the next term \(+38\): \[ -4x + 38 \]

Step 5: Repeat the process again

Now divide \(-4x\) by \(4x\): \[ \frac{-4x}{4x} = -1 \] Multiply \(-1\) by \(4x + 5\): \[ -1 \cdot (4x + 5) = -4x - 5 \] Now subtract this from \(-4x + 38\): \[ \begin{align*} (-4x + 38) - (-4x - 5) &= 43 \end{align*} \]

Step 6: Write the final result

Since the degree of the remainder \(43\) is less than the degree of the divisor \(4x + 5\), we have completed the division. The quotient is \(2x^2 - 2x - 1\) and the remainder is \(43\).

Final Answer

Expressing in the required form: \[ q(x) + \frac{r(x)}{b(x)} = 2x^2 - 2x - 1 + \frac{43}{4x + 5} \] This is the result of dividing \(8x^3 + 2x^2 - 14x + 38\) by \(4x + 5\).