Find the quotient and remainder using long division for $\frac{2 x^3-8 x^2+7 x-15}{2 x^2+5}$.

Question

kiahbreann04 kiahbreann04

04-11-2026
Mathematics

Answered

Find the quotient and remainder using long division for $\frac{2 x^3-8 x^2+7 x-15}{2 x^2+5}$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T01:36:33-05:00

Begin by dividing the lead term of the dividend, $2x^3$, by the lead term of the divisor, $2x^2$, yielding the quotient's first term, $x$.
Multiply the divisor by $x$ and subtract the product from the dividend, resulting in a new partial dividend: $-8x^2 + 2x - 15$.
Divide the new leading term $-8x^2$ by $2x^2$ to get the second quotient term, $-4$, then subtract $-4(2x^2 + 5)$ from the current partial dividend to obtain the final remainder, $2x + 5$.
The resulting quotient is $\boxed{x - 4}$ and the remainder is $\boxed{2x + 5}$.

Explanation

Problem Analysis

We are asked to find the quotient and remainder for the division of the polynomial $P(x) = 2x^3 - 8x^2 + 7x - 15$ by the polynomial $D(x) = 2x^2 + 5$. Using polynomial long division, we can express the dividend as $P(x) = D(x)Q(x) + R(x)$, where $Q(x)$ is the quotient and $R(x)$ is the remainder. The degree of the remainder $R(x)$ must be strictly less than the degree of the divisor $D(x)$. Since the divisor $D(x) = 2x^2 + 5$ has a degree of 2, the remainder $R(x)$ must be a polynomial of degree at most 1, of the form $ax + b$.

First Step of Division

To start the long division, we divide the leading term of the dividend ($2x^3$) by the leading term of the divisor ($2x^2$). This gives the first term of the quotient: $\frac{2x^3}{2x^2} = x$. Now, we multiply this term by the entire divisor: $x \cdot (2x^2 + 5) = 2x^3 + 5x$. Subtracting this from the original polynomial, we get: $(2x^3 - 8x^2 + 7x - 15) - (2x^3 + 5x) = -8x^2 + 2x - 15$.

Completion of Division

Next, we divide the leading term of the new partial dividend ($-8x^2$) by the leading term of the divisor ($2x^2$). This gives the second term of the quotient: $\frac{-8x^2}{2x^2} = -4$. Now, multiply this term by the entire divisor: $-4 \cdot (2x^2 + 5) = -8x^2 - 20$. Subtracting this result from the current partial dividend, we obtain: $(-8x^2 + 2x - 15) - (-8x^2 - 20) = 2x + (-15 + 20) = 2x + 5$. Since the degree of $2x + 5$ is 1, which is less than the degree of the divisor $2x^2 + 5$ (which is 2), the division process stops. Thus, the quotient is $x - 4$ and the remainder is $2x + 5$.

Examples

Polynomial long division is widely used in algebra and calculus, such as in performing partial fraction decomposition to simplify complicated integrals. For instance, if you were integrating a rational function, dividing the numerator by the denominator is the first step to break it down into simpler, integrable parts. Similarly, computer scientists use these algorithms in error-detecting codes and cryptography to process and verify data efficiently. Understanding how to manage polynomial division allows for a deeper grasp of how complex functions behave.