Use the long division method to find the result when $4 x^3+8 x^2+13 x+15$ is divided by $2 x+3$.

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03-11-2026
Mathematics

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Use the long division method to find the result when $4 x^3+8 x^2+13 x+15$ is divided by $2 x+3$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T08:18:23-05:00

Identify the dividend $4x^3 + 8x^2 + 13x + 15$ and divisor $2x + 3$.
Perform iterative division, multiplying and subtracting the leading terms at each step to determine the terms of the quotient.
Conclude that the quotient is $2x^2 + x + 5$ and the remainder is $0$.
The final result of the division is $\boxed{2x^2 + x + 5}$.

Explanation

Problem Analysis and Method Definition

The problem asks for the quotient and remainder when the polynomial $P(x) = 4x^3 + 8x^2 + 13x + 15$ is divided by the linear polynomial $D(x) = 2x + 3$. Polynomial long division is a systematic method similar to long division for integers, where we iteratively divide the leading term of the dividend by the leading term of the divisor to build the quotient. We expect a quotient of degree 2 and a remainder of degree 0 (or a constant), since the dividend is of degree 3 and the divisor is of degree 1.

First Step of Division

To perform the long division, we start by dividing the leading term of the dividend $4x^3$ by the leading term of the divisor $2x$:
\[\frac{4x^3}{2x} = 2x^2\]
This gives us the first term of our quotient. Next, we multiply this term by the entire divisor $2x + 3$:
\[2x^2(2x + 3) = 4x^3 + 6x^2\]
Now, we subtract this from the original dividend:
\[(4x^3 + 8x^2 + 13x + 15) - (4x^3 + 6x^2) = 2x^2 + 13x + 15\]
The resulting polynomial is $2x^2 + 13x + 15$.

Second Step of Division

Now we take the leading term of our new dividend $2x^2$ and divide it by the leading term of the divisor $2x$:
\[\frac{2x^2}{2x} = x\]
This is the second term of our quotient. We multiply this by the divisor $2x + 3$:
\[x(2x + 3) = 2x^2 + 3x\]
Subtracting this from our current dividend:
\[(2x^2 + 13x + 15) - (2x^2 + 3x) = 10x + 15\]
The resulting expression is $10x + 15$.

Final Step of Division and Conclusion

Finally, we divide the leading term of the remaining expression $10x$ by the leading term of the divisor $2x$:
\[\frac{10x}{2x} = 5\]
This gives the constant term of the quotient. We multiply this by the divisor:
\[5(2x + 3) = 10x + 15\]
Subtracting this from our current dividend:
\[(10x + 15) - (10x + 15) = 0\]
Since the subtraction results in zero, the remainder is $0$, and the division is complete. The quotient is the sum of the terms we calculated: $2x^2 + x + 5$. Thus, $4x^3 + 8x^2 + 13x + 15 = (2x + 3)(2x^2 + x + 5) + 0$. The final quotient is $2x^2 + x + 5$ and the remainder is $0$.

Examples

Polynomial long division is essential in engineering and physics, particularly when analyzing the stability of dynamical systems. For example, when solving differential equations, researchers often use the Laplace transform, which converts complex equations into algebraic ones involving rational functions (ratios of polynomials). If you have a fraction like $\frac{4s^3 + 8s^2 + 13s + 15}{2s+3}$, you can use long division to simplify it into a polynomial and a simpler fraction, allowing you to interpret the system's behavior, such as identifying natural frequencies or damping factors, more easily. This helps engineers design robust control systems for robotics or aerospace technology.