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(2x^(3)+4x^(2)+3x+1)÷(2x+1)

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

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Divide polynomials using long division

Full solution

Q.

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

Use polynomial long division: To divide the polynomial $(2x^3 + 4x^2 + 3x + 1)$ by the binomial $(2x + 1)$ , we will use polynomial long division.
Divide leading terms: First, we divide the leading term of the numerator, $2x^3$ , by the leading term of the denominator, $2x$ . This gives us $x^2$ . We then multiply the entire denominator $(2x + 1)$ by $x^2$ and subtract the result from the original polynomial.
Subtract and multiply: Multiplying $(2x + 1)$ by $x^2$ gives us $2x^3 + x^2$ . We write this below the original polynomial and subtract. $(2x^3 + 4x^2) - (2x^3 + x^2) = 3x^2$ .
Bring down next term: Bring down the next term of the original polynomial to get $3x^2 + 3x$ . Now, divide the leading term of this result, $3x^2$ , by the leading term of the denominator, $2x$ , to get $(\frac{3}{2})x$ .
Divide leading terms again: Multiply the entire denominator $(2x + 1)$ by $(\frac{3}{2})x$ and subtract the result from $(3x^2 + 3x)$ . Multiplying gives us $(3x^2 + (\frac{3}{2})x)$ , which we subtract from $(3x^2 + 3x)$ to get $(3x - (\frac{3}{2})x)$ .
Subtract and multiply: Simplifying $(3x - (\frac{3}{2})x)$ gives us $(\frac{3}{2})x$ . Bring down the next term of the original polynomial to get $(\frac{3}{2})x + 1$ . Now, divide the leading term of this result, $(\frac{3}{2})x$ , by the leading term of the denominator, $2x$ , to get $\frac{3}{4}$ .
Bring down next term again: Multiply the entire denominator $2x + 1$ by $\frac{3}{4}$ and subtract the result from $\left(\frac{3}{2}\right)x + 1$ . Multiplying gives us $\left(\frac{3}{2}\right)x + \frac{3}{4}$ , which we subtract from $\left(\frac{3}{2}\right)x + 1$ to get the remainder.
Divide leading terms once more: Simplifying $1 - \frac{3}{4}$ gives us $\frac{1}{4}$ . So, the remainder is $\frac{1}{4}$ , and the quotient we have obtained is $x^2 + \frac{3}{2}x + \frac{3}{4}$ .
Subtract and find remainder: The final answer is the quotient plus the remainder over the original divisor: $x^2 + \left(\frac{3}{2}\right)x + \frac{3}{4} + \frac{1}{4}\left(\frac{1}{2x + 1}\right)$ .

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