Solved on Mar 05, 2024

Divide $4x^3 - 16x^2 + 18x - 28$ by $-x + 3$ using long division. Find the quotient and remainder.

STEP 1

Assumptions
1. We are dividing a polynomial by a binomial using long division.
2. The polynomial to be divided is $4x^3 - 16x^2 + 18x - 28$ .
3. The binomial divisor is $-x + 3$ .

STEP 2

Set up the long division by writing the polynomial (dividend) inside the division symbol and the binomial (divisor) outside.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \end{array}$

STEP 3

Divide the first term of the polynomial by the first term of the binomial to get the first term of the quotient.
$\frac{4x^3}{-x} = -4x^2$

STEP 4

Write the first term of the quotient above the division symbol.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 \\ \end{array}$

STEP 5

Multiply the entire binomial by the first term of the quotient and write the result under the corresponding terms of the polynomial.
$(-x+3)(-4x^2) = 4x^3 - 12x^2$
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 \\ & -(4x^3 - 12x^2) \\ \end{array}$

STEP 6

Subtract the result from the polynomial (change the signs and add).
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & 0 - 4x^2 \\ \end{array}$

STEP 7

Bring down the next term of the polynomial to continue the division.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ \end{array}$

STEP 8

Divide the new first term of the remaining polynomial by the first term of the binomial to get the next term of the quotient.
$\frac{-4x^2}{-x} = 4x$

STEP 9

Write the next term of the quotient above the division symbol.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ \end{array}$

STEP 10

Multiply the binomial by the new term of the quotient and write the result under the corresponding terms of the polynomial.
$(-x+3)(4x) = -4x^2 + 12x$
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ & -(-4x^2 + 12x) \\ \end{array}$

STEP 11

Subtract the result from the remaining polynomial.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ & -(-4x^2 + 12x) \\ & \underline{-4x^2 + 18x} \\ & 0 + 6x \\ \end{array}$

STEP 12

Bring down the next term of the polynomial to continue the division.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ & -(-4x^2 + 12x) \\ & \underline{-4x^2 + 18x} \\ & 6x - 28 \\ \end{array}$

STEP 13

Divide the new first term of the remaining polynomial by the first term of the binomial to get the next term of the quotient.
$\frac{6x}{-x} = -6$

STEP 14

Write the next term of the quotient above the division symbol.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x - 6 \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ & -(-4x^2 + 12x) \\ & \underline{-4x^2 + 18x} \\ & 6x - 28 \\ \end{array}$

STEP 15

Multiply the binomial by the new term of the quotient and write the result under the corresponding terms of the polynomial.
$(-x+3)(-6) = 6x - 18$
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x - 6 \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ & -(-4x^2 + 12x) \\ & \underline{-4x^2 + 18x} \\ & 6x - 28 \\ & -(6x - 18) \\ \end{array}$

STEP 16

Subtract the result from the remaining polynomial.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x - 6 \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ & -(-4x^2 + 12x) \\ & \underline{-4x^2 + 18x} \\ & 6x - 28 \\ & -(6x - 18) \\ & \underline{6x - 28} \\ & 0 - 10 \\ \end{array}$

STEP 17

The remainder is $-10$ , which cannot be divided further by the binomial.

STEP 18

Write the final quotient and remainder.
The quotient is $-4x^2 + 4x - 6$ and the remainder is $-10$ .

STEP 19

Choose the correct answer from the given options.
The correct answer is C) $-4x^2 + 4x - 6 \quad R(-10)$ .

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Divide $4x^3 - 16x^2 + 18x - 28$ by $-x + 3$ using long division. Find the quotient and remainder.

STEP 1

Assumptions
1. We are dividing a polynomial by a binomial using long division.
2. The polynomial to be divided is $4x^3 - 16x^2 + 18x - 28$ .
3. The binomial divisor is $-x + 3$ .

STEP 2

Set up the long division by writing the polynomial (dividend) inside the division symbol and the binomial (divisor) outside.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \end{array}$

STEP 3

Divide the first term of the polynomial by the first term of the binomial to get the first term of the quotient.
$\frac{4x^3}{-x} = -4x^2$

STEP 4

Write the first term of the quotient above the division symbol.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 \\ \end{array}$

STEP 5

Multiply the entire binomial by the first term of the quotient and write the result under the corresponding terms of the polynomial.
$(-x+3)(-4x^2) = 4x^3 - 12x^2$
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 \\ & -(4x^3 - 12x^2) \\ \end{array}$

STEP 6

Subtract the result from the polynomial (change the signs and add).
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & 0 - 4x^2 \\ \end{array}$

STEP 7

Bring down the next term of the polynomial to continue the division.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ \end{array}$

STEP 8

Divide the new first term of the remaining polynomial by the first term of the binomial to get the next term of the quotient.
$\frac{-4x^2}{-x} = 4x$

STEP 9

Write the next term of the quotient above the division symbol.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ \end{array}$

STEP 10

STEP 11

Subtract the result from the remaining polynomial.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ & -(-4x^2 + 12x) \\ & \underline{-4x^2 + 18x} \\ & 0 + 6x \\ \end{array}$

STEP 12

Bring down the next term of the polynomial to continue the division.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ & -(-4x^2 + 12x) \\ & \underline{-4x^2 + 18x} \\ & 6x - 28 \\ \end{array}$

STEP 13

Divide the new first term of the remaining polynomial by the first term of the binomial to get the next term of the quotient.
$\frac{6x}{-x} = -6$

STEP 14

Write the next term of the quotient above the division symbol.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 + 4x - 6 \\ & -(4x^3 - 12x^2) \\ & \underline{4x^3 - 16x^2} \\ & -4x^2 + 18x \\ & -(-4x^2 + 12x) \\ & \underline{-4x^2 + 18x} \\ & 6x - 28 \\ \end{array}$

STEP 15

STEP 16

STEP 17

The remainder is $-10$ , which cannot be divided further by the binomial.

STEP 18

Write the final quotient and remainder.
The quotient is $-4x^2 + 4x - 6$ and the remainder is $-10$ .

STEP 19

Choose the correct answer from the given options.
The correct answer is C) $-4x^2 + 4x - 6 \quad R(-10)$ .

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Divide 4x3−16x2+18x−284x^3 - 16x^2 + 18x - 284x3−16x2+18x−28 by −x+3-x + 3−x+3 using long division. Find the quotient and remainder.

STEP 1

Assumptions1. We are dividing a polynomial by a binomial using long division.2. The polynomial to be divided is 4x3−16x2+18x−284x^3 - 16x^2 + 18x - 284x3−16x2+18x−28.3. The binomial divisor is −x+3-x + 3−x+3.

STEP 2

Set up the long division by writing the polynomial (dividend) inside the division symbol and the binomial (divisor) outside.−x+34x3−16x2+18x−28\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \end{array}−x+3​4x3−16x2+18x−28​

STEP 3

Divide the first term of the polynomial by the first term of the binomial to get the first term of the quotient.4x3−x=−4x2\frac{4x^3}{-x} = -4x^2−x4x3​=−4x2

STEP 4

Write the first term of the quotient above the division symbol.−x+34x3−16x2+18x−28−4x2\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 \\ \end{array}−x+3​4x3−16x2+18x−28−4x2​​

STEP 5

STEP 6

STEP 7

STEP 8

Divide the new first term of the remaining polynomial by the first term of the binomial to get the next term of the quotient.−4x2−x=4x\frac{-4x^2}{-x} = 4x−x−4x2​=4x

STEP 9

STEP 10

STEP 11

STEP 12

STEP 13

Divide the new first term of the remaining polynomial by the first term of the binomial to get the next term of the quotient.6x−x=−6\frac{6x}{-x} = -6−x6x​=−6

STEP 14

STEP 15

STEP 16

STEP 17

The remainder is −10-10−10, which cannot be divided further by the binomial.

STEP 18

Write the final quotient and remainder.The quotient is −4x2+4x−6-4x^2 + 4x - 6−4x2+4x−6 and the remainder is −10-10−10.

STEP 19

Choose the correct answer from the given options.The correct answer is C) −4x2+4x−6R(−10)-4x^2 + 4x - 6 \quad R(-10)−4x2+4x−6R(−10).

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Divide $4x^3 - 16x^2 + 18x - 28$ by $-x + 3$ using long division. Find the quotient and remainder.

Assumptions
1. We are dividing a polynomial by a binomial using long division.
2. The polynomial to be divided is $4x^3 - 16x^2 + 18x - 28$ .
3. The binomial divisor is $-x + 3$ .

Set up the long division by writing the polynomial (dividend) inside the division symbol and the binomial (divisor) outside.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \end{array}$

Divide the first term of the polynomial by the first term of the binomial to get the first term of the quotient.
$\frac{4x^3}{-x} = -4x^2$

Write the first term of the quotient above the division symbol.
$\begin{array}{r|l} -x+3 & 4x^3 - 16x^2 + 18x - 28 \\ \hline & -4x^2 \\ \end{array}$

Divide the new first term of the remaining polynomial by the first term of the binomial to get the next term of the quotient.
$\frac{-4x^2}{-x} = 4x$

Divide the new first term of the remaining polynomial by the first term of the binomial to get the next term of the quotient.
$\frac{6x}{-x} = -6$

The remainder is $-10$ , which cannot be divided further by the binomial.

Write the final quotient and remainder.
The quotient is $-4x^2 + 4x - 6$ and the remainder is $-10$ .

Choose the correct answer from the given options.
The correct answer is C) $-4x^2 + 4x - 6 \quad R(-10)$ .