Find the Remainder (x^4-6x^2+8)÷(x- square root of 2)

$(x^{4} - 6 x^{2} + 8) \div (x - \sqrt{2})$

Step 1

To calculate the remainder, first divide the polynomials.

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - x^{3} \sqrt{2}$

$x^{3}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

Divide the highest order term in the dividend $x^{3} \sqrt{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2} \sqrt{2}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2} \sqrt{2}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} \sqrt{2} - 2 x^{2}$

$x^{3}$

$x^{2} \sqrt{2}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x^{2} \sqrt{2}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x^{2} \sqrt{2}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

$0 x$

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2} \sqrt{2}$

$4 x$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

$0 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2} \sqrt{2}$

$4 x$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x \sqrt{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 4 x \sqrt{2}$

$x^{3}$

$x^{2} \sqrt{2}$

$4 x$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x \sqrt{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x^{2} \sqrt{2}$

$4 x$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x \sqrt{2}$

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x^{2} \sqrt{2}$

$4 x$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x \sqrt{2}$

$8$

Divide the highest order term in the dividend $- 4 x \sqrt{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2} \sqrt{2}$

$4 x$

$4 \sqrt{2}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x \sqrt{2}$

$8$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2} \sqrt{2}$

$4 x$

$4 \sqrt{2}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x \sqrt{2}$

$8$

$4 \sqrt{2} x$

$8$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 \sqrt{2} x + 8$

$x^{3}$

$x^{2} \sqrt{2}$

$4 x$

$4 \sqrt{2}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x \sqrt{2}$

$8$

$4 \sqrt{2} x$

$8$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x^{2} \sqrt{2}$

$4 x$

$4 \sqrt{2}$

$x$

$\sqrt{2}$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$8$

$x^{4}$

$x^{3} \sqrt{2}$

$6 x^{2}$

$x^{3} \sqrt{2}$

$2 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x \sqrt{2}$

$8$

$4 \sqrt{2} x$

$8$

$0$

Since the remander is $0$ , the final answer is the quotient.

$x^{3} + x^{2} \sqrt{2} - 4 x - 4 \sqrt{2}$

Step 2

Since the final term in the resulting expression is not a fraction, the remainder is $0$ .

$0$

Find the Remainder (x^4-6x^2+8)÷(x- square root of 2)

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