Math Snap

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:
$$(248x) - (248x - 992) = 992$$

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:
$$\frac{4x^4 - 3x^2 + 4x}{x - 4} = 4x^3 + 16x^2 + 61x + 248 + \frac{992}{x - 4}$$