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Use long division to find the quotient and the remainder.
(2x^(3)+4x^(2)-2)÷(x^(2)+5x+1)

$9$ ) Use long division to find the quotient and the remainder. $\left(2 x^{3}+4 x^{2}-2\right) \div\left(x^{2}+5 x+1\right)$

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Multiply and divide rational expressions

Full solution

Q.

9

) Use long division to find the quotient and the remainder.

\left(2 x^{3}+4 x^{2}-2\right) \div\left(x^{2}+5 x+1\right)

Set up division: First, set up the long division by writing $(2x^3 + 4x^2 + 0x - 2)$ inside the division bracket and $(x^2 + 5x + 1)$ outside.
Find first quotient term: Divide the first term of the dividend, $2x^3$ , by the first term of the divisor, $x^2$ , to get the first term of the quotient, which is $2x$ .
Multiply and subtract: Multiply the entire divisor $(x^2 + 5x + 1)$ by the first term of the quotient $(2x)$ to get $(2x^3 + 10x^2 + 2x)$ .
Find second quotient term: Subtract this from the dividend: $2x^3 + 4x^2 + 0x - 2$ - $2x^3 + 10x^2 + 2x$ to get the new dividend $-6x^2 - 2x - 2$ .
Multiply and subtract: Divide the first term of the new dividend, $-6x^2$ , by the first term of the divisor, $x^2$ , to get the second term of the quotient, which is $-6$ .
Check for further division: Multiply the entire divisor $x^2 + 5x + 1$ by the second term of the quotient $-6$ to get $-6x^2 - 30x - 6$ .
Check for further division: Multiply the entire divisor $x^2 + 5x + 1$ by the second term of the quotient $-6$ to get $-6x^2 - 30x - 6$ . Subtract this from the new dividend: $(-6x^2 - 2x - 2) - (-6x^2 - 30x - 6)$ to get the new dividend $28x + 4$ .
Check for further division: Multiply the entire divisor $x^2 + 5x + 1$ by the second term of the quotient $-6$ to get $-6x^2 - 30x - 6$ . Subtract this from the new dividend: $(-6x^2 - 2x - 2) - (-6x^2 - 30x - 6)$ to get the new dividend $28x + 4$ . Since the degree of the new dividend $28x + 4$ is less than the degree of the divisor $x^2 + 5x + 1$ , we cannot divide further, and this becomes our remainder.

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