Math / Algebra

QuestionIn the following problem, divide using long division. State the quotient, $\mathrm{q}(\mathrm{x})$ , and the remainder, $\mathrm{r}(\mathrm{x})$ . $\begin{array}{l} \frac{4 x^{4}-3 x^{2}+4 x}{x-4} \\ \frac{4 x^{4}-3 x^{2}+4 x}{x-4}=\square+\frac{\square}{x-4} \end{array}$ (Simplify your answers. Do not factor. Use integers or fractions for any numbers in the expressions.)

Studdy Solution

STEP 1

1. We are performing polynomial long division.
2. The dividend is $4x^4 - 3x^2 + 4x$ .
3. The divisor is $x - 4$ .
4. We need to find both the quotient and the remainder.

STEP 2

1. Set up the long division.
2. Perform the division step-by-step.
3. Identify the quotient and remainder.

STEP 3

Set up the polynomial long division by writing the dividend $4x^4 - 3x^2 + 4x$ under the division symbol and the divisor $x - 4$ outside.
$\text{Divide } 4x^4 \text{ by } x \text{ to get the first term of the quotient: } 4x^3$

STEP 4

Multiply $4x^3$ by the divisor $x - 4$ and subtract the result from the dividend.
$4x^3 \times (x - 4) = 4x^4 - 16x^3$
Subtract:
$(4x^4 - 3x^2 + 4x) - (4x^4 - 16x^3) = 16x^3 - 3x^2 + 4x$

STEP 5

Divide $16x^3$ by $x$ to get the next term of the quotient: $16x^2$ .
Multiply $16x^2$ by $x - 4$ and subtract the result from the current dividend.
$16x^2 \times (x - 4) = 16x^3 - 64x^2$
Subtract:
$(16x^3 - 3x^2 + 4x) - (16x^3 - 64x^2) = 61x^2 + 4x$

STEP 6

Divide $61x^2$ by $x$ to get the next term of the quotient: $61x$ .
Multiply $61x$ by $x - 4$ and subtract the result from the current dividend.
$61x \times (x - 4) = 61x^2 - 244x$
Subtract:
$(61x^2 + 4x) - (61x^2 - 244x) = 248x$

STEP 7

Divide $248x$ by $x$ to get the next term of the quotient: $248$ .
Multiply $248$ by $x - 4$ and subtract the result from the current dividend.
$248 \times (x - 4) = 248x - 992$
Subtract:
$(248x) - (248x - 992) = 992$

STEP 8

Since there are no more terms to bring down, the remainder is $992$ .
The quotient is $4x^3 + 16x^2 + 61x + 248$ and the remainder is $992$ .
The final result of the division is:
$\frac{4x^4 - 3x^2 + 4x}{x - 4} = 4x^3 + 16x^2 + 61x + 248 + \frac{992}{x - 4}$

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Studdy Solution

STEP 1

1. We are performing polynomial long division.
2. The dividend is $4x^4 - 3x^2 + 4x$ .
3. The divisor is $x - 4$ .
4. We need to find both the quotient and the remainder.

STEP 2

1. Set up the long division.
2. Perform the division step-by-step.
3. Identify the quotient and remainder.

STEP 3

Set up the polynomial long division by writing the dividend $4x^4 - 3x^2 + 4x$ under the division symbol and the divisor $x - 4$ outside.
$\text{Divide } 4x^4 \text{ by } x \text{ to get the first term of the quotient: } 4x^3$

STEP 4

Multiply $4x^3$ by the divisor $x - 4$ and subtract the result from the dividend.
$4x^3 \times (x - 4) = 4x^4 - 16x^3$
Subtract:
$(4x^4 - 3x^2 + 4x) - (4x^4 - 16x^3) = 16x^3 - 3x^2 + 4x$

STEP 5

Divide $16x^3$ by $x$ to get the next term of the quotient: $16x^2$ .
Multiply $16x^2$ by $x - 4$ and subtract the result from the current dividend.
$16x^2 \times (x - 4) = 16x^3 - 64x^2$
Subtract:
$(16x^3 - 3x^2 + 4x) - (16x^3 - 64x^2) = 61x^2 + 4x$

STEP 6

Divide $61x^2$ by $x$ to get the next term of the quotient: $61x$ .
Multiply $61x$ by $x - 4$ and subtract the result from the current dividend.
$61x \times (x - 4) = 61x^2 - 244x$
Subtract:
$(61x^2 + 4x) - (61x^2 - 244x) = 248x$

STEP 7

Divide $248x$ by $x$ to get the next term of the quotient: $248$ .
Multiply $248$ by $x - 4$ and subtract the result from the current dividend.
$248 \times (x - 4) = 248x - 992$
Subtract:
$(248x) - (248x - 992) = 992$

STEP 8

Since there are no more terms to bring down, the remainder is $992$ .
The quotient is $4x^3 + 16x^2 + 61x + 248$ and the remainder is $992$ .
The final result of the division is:
$\frac{4x^4 - 3x^2 + 4x}{x - 4} = 4x^3 + 16x^2 + 61x + 248 + \frac{992}{x - 4}$

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Studdy Solution

STEP 1

1. We are performing polynomial long division.2. The dividend is 4x4−3x2+4x 4x^4 - 3x^2 + 4x 4x4−3x2+4x.3. The divisor is x−4 x - 4 x−4.4. We need to find both the quotient and the remainder.

STEP 2

1. Set up the long division.2. Perform the division step-by-step.3. Identify the quotient and remainder.

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

STEP 8

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. We are performing polynomial long division.
2. The dividend is $4x^4 - 3x^2 + 4x$ .
3. The divisor is $x - 4$ .
4. We need to find both the quotient and the remainder.

1. Set up the long division.
2. Perform the division step-by-step.
3. Identify the quotient and remainder.