$x + 1 \longdiv { x ^ { 3 } - 4 x ^ { 2 } + x - 2 }$

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06-11-2026
Mathematics

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$x + 1 \longdiv { x ^ { 3 } - 4 x ^ { 2 } + x - 2 }$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T13:17:45-05:00

Identify coefficients of the dividend $P(x) = x^3 - 4x^2 + x - 2$ as $1, -4, 1, -2$ and the root of the divisor $x+1=0$ as $c = -1$.
Execute synthetic division to obtain the quotient coefficients $1, -5, 6$ and remainder $-8$.
Construct the quotient polynomial $Q(x) = x^2 - 5x + 6$.
The final result is: quotient $Q(x) = x^2 - 5x + 6$ and remainder $R = \boxed{-8}$.

Explanation

Understanding the Problem and Data Analysis

We are tasked with dividing the cubic polynomial $P(x) = x^3 - 4x^2 + x - 2$ by the linear polynomial $D(x) = x + 1$. The goal is to express the dividend in the form $P(x) = D(x)Q(x) + R$, where $Q(x)$ is the quotient and $R$ is the remainder. Since the divisor is a first-degree polynomial, the remainder $R$ must be a constant value. We can determine the quotient and remainder using either polynomial long division or synthetic division.

Performing Polynomial Division

Using synthetic division, we identify the root of the divisor $x+1=0$, which is $c = -1$. The coefficients of the dividend $x^3 - 4x^2 + x - 2$ are $1, -4, 1,$ and $-2$.

We set up the synthetic division tableau:

Bring down the first coefficient: $1$.
Multiply by $-1$ to get $-1$, add to the second coefficient $-4$ to get $-5$.
Multiply $-5$ by $-1$ to get $5$, add to the third coefficient $1$ to get $6$.
Multiply $6$ by $-1$ to get $-6$, add to the constant term $-2$ to get $-8$.

The resulting coefficients $1, -5, 6$ correspond to the quadratic quotient $Q(x) = x^2 - 5x + 6$, and the final result $-8$ is the remainder $R$.

Verification of the Result

To verify our results, we check if $(x+1)(x^2 - 5x + 6) - 8$ equals the original dividend $P(x)$.
Expanding the product:
$(x+1)(x^2 - 5x + 6) = x(x^2 - 5x + 6) + 1(x^2 - 5x + 6)$
$= x^3 - 5x^2 + 6x + x^2 - 5x + 6$
$= x^3 - 4x^2 + x + 6$

Now, adding the remainder $R = -8$:
$x^3 - 4x^2 + x + 6 - 8 = x^3 - 4x^2 + x - 2$
This confirms our quotient $Q(x) = x^2 - 5x + 6$ and remainder $R = -8$ are correct.

Conclusion

The division of $x^3 - 4x^2 + x - 2$ by $x + 1$ yields a quotient of $x^2 - 5x + 6$ and a remainder of $-8$. We can write the result as:
\[ \frac{x^3 - 4x^2 + x - 2}{x + 1} = x^2 - 5x + 6 - \frac{8}{x + 1} \]
Thus, the quotient is $x^2 - 5x + 6$ and the remainder is $-8$.

Examples

Polynomial division is a fundamental tool in algebra and calculus, especially when simplifying rational expressions or finding roots of higher-degree polynomials. For example, if you are designing a computer graphics algorithm to determine the intersection points of complex curves defined by polynomials, you often need to perform division to reduce the degree of these equations. This allows you to break down complicated curves into simpler, manageable components, ensuring that your rendering engine can calculate coordinates quickly and accurately.