\left(x^{3}+x^{2}-36x+42\right)\div\left(x+7\right)

To simplify the expression \((x^3 + x^2 - 36x + 42) \div (x + 7)\), we will perform polynomial long division. Alternatively, we could use synthetic division since the divisor is a linear polynomial, but for clarity, let's proceed with polynomial long division.

### Step-by-Step Polynomial Long Division

#### Step 1: Set Up the Division

The dividend is:
\[
x^3 + x^2 - 36x + 42
\]

The divisor is:
\[
x + 7
\]

We set up the division as follows:

\[
\begin{array}{r|l}
x + 7 & x^3 + x^2 - 36x + 42 \\
\end{array}
\]

#### Step 2: Divide the Leading Terms

Divide the leading term of the dividend by the leading term of the divisor:

\[
\frac{x^3}{x} = x^2
\]

Place \(x^2\) above the division bar.

#### Step 3: Multiply and Subtract

Multiply \(x^2\) by the divisor \((x + 7)\):

\[
x^2 \cdot (x + 7) = x^3 + 7x^2
\]

Subtract this from the original dividend:

\[
(x^3 + x^2 - 36x + 42) - (x^3 + 7x^2) = -6x^2 - 36x + 42
\]

#### Step 4: Repeat the Process

Divide the new leading term by the leading term of the divisor:

\[
\frac{-6x^2}{x} = -6x
\]

Place \(-6x\) above the division bar.

Multiply \(-6x\) by the divisor \((x + 7)\):

\[
-6x \cdot (x + 7) = -6x^2 - 42x
\]

Subtract this from the current result:

\[
(-6x^2 - 36x + 42) - (-6x^2 - 42x) = 6x + 42
\]

#### Step 5: Divide Again

Divide the new leading term by the leading term of the divisor:

\[
\frac{6x}{x} = 6
\]

Place \(6\) above the division bar.

Multiply \(6\) by the divisor \((x + 7)\):

\[
6 \cdot (x + 7) = 6x + 42
\]

Subtract this from the current result:

\[
(6x + 42) - (6x + 42) = 0
\]

#### Step 6: Conclusion

Since the remainder is \(0\), the division is exact. The quotient is:

\[
x^2 - 6x + 6
\]

### Final Answer

\[
(x^3 + x^2 - 36x + 42) \div (x + 7) = x^2 - 6x + 6
\]

### Would you like further details or have any questions?

#### Related Questions:
1. How can synthetic division be used to divide polynomials?
2. What are the advantages of polynomial long division over synthetic division?
3. How do you determine if a divisor is a factor of a polynomial?
4. What is the Factor Theorem, and how does it relate to polynomial division?
5. How can you use polynomial division to simplify rational expressions?

#### Tip:
When performing polynomial long division, carefully align like terms and ensure each step of multiplication and subtraction is accurate to avoid errors.

Learn how to simplify the expression (x^3 + x^2 - 36x + 42) divided by (x + 7) using polynomial long division. This detailed step-by-step solution covers key algebraic concepts in polynomial division. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation