Solved on Feb 07, 2024

Rewrite $12x^2 - 36x + 28$ in the form $(4x - 4) \times q(x) + r(x)$ .

STEP 1

Assumptions
1. We have a polynomial $12x^2 - 36x + 28$ .
2. We want to rewrite this polynomial in the form $(4x - 4) \times q(x) + r(x)$ .
3. $q(x)$ is the quotient polynomial when the original polynomial is divided by $(4x - 4)$ .
4. $r(x)$ is the remainder polynomial when the original polynomial is divided by $(4x - 4)$ .

STEP 2

To rewrite the polynomial, we will perform polynomial long division, dividing $12x^2 - 36x + 28$ by $(4x - 4)$ .

STEP 3

Set up the division:
$\begin{array}{r} q(x) \\ 4x - 4 \,|\overline{\,12x^2 - 36x + 28} \\ \end{array}$

STEP 4

Determine the first term of $q(x)$ by dividing the leading term of the polynomial $12x^2$ by the leading term of $4x - 4$ , which is $4x$ .
$\frac{12x^2}{4x} = 3x$

STEP 5

Multiply $(4x - 4)$ by the first term of $q(x)$ , which is $3x$ .
$(4x - 4) \times 3x = 12x^2 - 12x$

STEP 6

Subtract this result from the original polynomial:
$\begin{array}{r} 12x^2 - 36x + 28 \\ -(12x^2 - 12x) \\ \hline -24x + 28 \\ \end{array}$

STEP 7

Determine the next term of $q(x)$ by dividing the new leading term of the polynomial $-24x$ by the leading term of $4x - 4$ , which is $4x$ .
$\frac{-24x}{4x} = -6$

STEP 8

Multiply $(4x - 4)$ by the next term of $q(x)$ , which is $-6$ .
$(4x - 4) \times -6 = -24x + 24$

STEP 9

Subtract this result from the polynomial we obtained after the first subtraction:
$\begin{array}{r} -24x + 28 \\ -(-24x + 24) \\ \hline 4 \\ \end{array}$

STEP 10

Now we have the remainder, which is $4$ . Since the degree of the remainder is less than the degree of the divisor $(4x - 4)$ , we can stop the division process.

STEP 11

Combine the terms of $q(x)$ and the remainder $r(x)$ to rewrite the original polynomial in the desired form:
$12x^2 - 36x + 28 = (4x - 4) \times (3x - 6) + 4$

STEP 12

Verify the result by expanding $(4x - 4) \times (3x - 6) + 4$ :
$(4x - 4) \times (3x - 6) + 4 = (12x^2 - 12x) + (-24x + 24) + 4$
$= 12x^2 - 12x - 24x + 24 + 4$
$= 12x^2 - 36x + 28$

STEP 13

Since the expansion matches the original polynomial, our division and rewriting are correct.
The polynomial $12x^2 - 36x + 28$ rewritten in the form $(4x - 4) \times q(x) + r(x)$ is:
$12x^2 - 36x + 28 = (4x - 4) \times (3x - 6) + 4$

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Rewrite $12x^2 - 36x + 28$ in the form $(4x - 4) \times q(x) + r(x)$ .

STEP 1

STEP 2

To rewrite the polynomial, we will perform polynomial long division, dividing $12x^2 - 36x + 28$ by $(4x - 4)$ .

STEP 3

Set up the division:
$\begin{array}{r} q(x) \\ 4x - 4 \,|\overline{\,12x^2 - 36x + 28} \\ \end{array}$

STEP 4

Determine the first term of $q(x)$ by dividing the leading term of the polynomial $12x^2$ by the leading term of $4x - 4$ , which is $4x$ .
$\frac{12x^2}{4x} = 3x$

STEP 5

Multiply $(4x - 4)$ by the first term of $q(x)$ , which is $3x$ .
$(4x - 4) \times 3x = 12x^2 - 12x$

STEP 6

Subtract this result from the original polynomial:
$\begin{array}{r} 12x^2 - 36x + 28 \\ -(12x^2 - 12x) \\ \hline -24x + 28 \\ \end{array}$

STEP 7

Determine the next term of $q(x)$ by dividing the new leading term of the polynomial $-24x$ by the leading term of $4x - 4$ , which is $4x$ .
$\frac{-24x}{4x} = -6$

STEP 8

Multiply $(4x - 4)$ by the next term of $q(x)$ , which is $-6$ .
$(4x - 4) \times -6 = -24x + 24$

STEP 9

Subtract this result from the polynomial we obtained after the first subtraction:
$\begin{array}{r} -24x + 28 \\ -(-24x + 24) \\ \hline 4 \\ \end{array}$

STEP 10

Now we have the remainder, which is $4$ . Since the degree of the remainder is less than the degree of the divisor $(4x - 4)$ , we can stop the division process.

STEP 11

Combine the terms of $q(x)$ and the remainder $r(x)$ to rewrite the original polynomial in the desired form:
$12x^2 - 36x + 28 = (4x - 4) \times (3x - 6) + 4$

STEP 12

Verify the result by expanding $(4x - 4) \times (3x - 6) + 4$ :
$(4x - 4) \times (3x - 6) + 4 = (12x^2 - 12x) + (-24x + 24) + 4$
$= 12x^2 - 12x - 24x + 24 + 4$
$= 12x^2 - 36x + 28$

STEP 13

Since the expansion matches the original polynomial, our division and rewriting are correct.
The polynomial $12x^2 - 36x + 28$ rewritten in the form $(4x - 4) \times q(x) + r(x)$ is:
$12x^2 - 36x + 28 = (4x - 4) \times (3x - 6) + 4$

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Rewrite 12x2−36x+2812x^2 - 36x + 2812x2−36x+28 in the form (4x−4)×q(x)+r(x)(4x - 4) \times q(x) + r(x)(4x−4)×q(x)+r(x).

STEP 1

STEP 2

To rewrite the polynomial, we will perform polynomial long division, dividing 12x2−36x+2812x^2 - 36x + 2812x2−36x+28 by (4x−4)(4x - 4)(4x−4).

STEP 3

Set up the division:q(x)4x−4 ∣ 12x2−36x+28‾ \begin{array}{r} q(x) \\ 4x - 4 \,|\overline{\,12x^2 - 36x + 28} \\ \end{array} q(x)4x−4∣12x2−36x+28​​

STEP 4

Determine the first term of q(x)q(x)q(x) by dividing the leading term of the polynomial 12x212x^212x2 by the leading term of 4x−44x - 44x−4, which is 4x4x4x.12x24x=3x \frac{12x^2}{4x} = 3x 4x12x2​=3x

STEP 5

Multiply (4x−4)(4x - 4)(4x−4) by the first term of q(x)q(x)q(x), which is 3x3x3x.(4x−4)×3x=12x2−12x (4x - 4) \times 3x = 12x^2 - 12x (4x−4)×3x=12x2−12x

STEP 6

Subtract this result from the original polynomial:12x2−36x+28−(12x2−12x)−24x+28 \begin{array}{r} 12x^2 - 36x + 28 \\ -(12x^2 - 12x) \\ \hline -24x + 28 \\ \end{array} 12x2−36x+28−(12x2−12x)−24x+28​​

STEP 7

Determine the next term of q(x)q(x)q(x) by dividing the new leading term of the polynomial −24x-24x−24x by the leading term of 4x−44x - 44x−4, which is 4x4x4x.−24x4x=−6 \frac{-24x}{4x} = -6 4x−24x​=−6

STEP 8

Multiply (4x−4)(4x - 4)(4x−4) by the next term of q(x)q(x)q(x), which is −6-6−6.(4x−4)×−6=−24x+24 (4x - 4) \times -6 = -24x + 24 (4x−4)×−6=−24x+24

STEP 9

Subtract this result from the polynomial we obtained after the first subtraction:−24x+28−(−24x+24)4 \begin{array}{r} -24x + 28 \\ -(-24x + 24) \\ \hline 4 \\ \end{array} −24x+28−(−24x+24)4​​

STEP 10

Now we have the remainder, which is 444. Since the degree of the remainder is less than the degree of the divisor (4x−4)(4x - 4)(4x−4), we can stop the division process.

STEP 11

Combine the terms of q(x)q(x)q(x) and the remainder r(x)r(x)r(x) to rewrite the original polynomial in the desired form:12x2−36x+28=(4x−4)×(3x−6)+4 12x^2 - 36x + 28 = (4x - 4) \times (3x - 6) + 4 12x2−36x+28=(4x−4)×(3x−6)+4

STEP 12

STEP 13

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Rewrite $12x^2 - 36x + 28$ in the form $(4x - 4) \times q(x) + r(x)$ .

To rewrite the polynomial, we will perform polynomial long division, dividing $12x^2 - 36x + 28$ by $(4x - 4)$ .

Set up the division:
$\begin{array}{r} q(x) \\ 4x - 4 \,|\overline{\,12x^2 - 36x + 28} \\ \end{array}$

Determine the first term of $q(x)$ by dividing the leading term of the polynomial $12x^2$ by the leading term of $4x - 4$ , which is $4x$ .
$\frac{12x^2}{4x} = 3x$

Multiply $(4x - 4)$ by the first term of $q(x)$ , which is $3x$ .
$(4x - 4) \times 3x = 12x^2 - 12x$

Subtract this result from the original polynomial:
$\begin{array}{r} 12x^2 - 36x + 28 \\ -(12x^2 - 12x) \\ \hline -24x + 28 \\ \end{array}$

Determine the next term of $q(x)$ by dividing the new leading term of the polynomial $-24x$ by the leading term of $4x - 4$ , which is $4x$ .
$\frac{-24x}{4x} = -6$

Multiply $(4x - 4)$ by the next term of $q(x)$ , which is $-6$ .
$(4x - 4) \times -6 = -24x + 24$

Subtract this result from the polynomial we obtained after the first subtraction:
$\begin{array}{r} -24x + 28 \\ -(-24x + 24) \\ \hline 4 \\ \end{array}$

Now we have the remainder, which is $4$ . Since the degree of the remainder is less than the degree of the divisor $(4x - 4)$ , we can stop the division process.

Combine the terms of $q(x)$ and the remainder $r(x)$ to rewrite the original polynomial in the desired form:
$12x^2 - 36x + 28 = (4x - 4) \times (3x - 6) + 4$