Use the long division method to find the result when 3, x, cubed, plus, 13, x, squared, plus, 5, x, minus, 143x 
3
 +13x 
2
 +5x−14 is divided by x, plus, 1x+1. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ 
b(x)
r(x)
​
 .

To perform the long division of the polynomial \( 3x^3 + 13x^2 + 5x - 14 \) by \( x + 1 \), follow these steps:

1. **Step 1: Divide the leading term of the dividend by the leading term of the divisor.**

   The leading term of the dividend is \( 3x^3 \), and the leading term of the divisor is \( x \). Divide \( 3x^3 \) by \( x \), which gives \( 3x^2 \).

   Now, multiply \( 3x^2 \) by \( x + 1 \):
   \[
   3x^2 \times (x + 1) = 3x^3 + 3x^2
   \]
   Subtract this from the original polynomial:
   \[
   (3x^3 + 13x^2 + 5x - 14) - (3x^3 + 3x^2) = 10x^2 + 5x - 14
   \]

2. **Step 2: Divide the new leading term of the dividend by the leading term of the divisor.**

   Now, the new leading term is \( 10x^2 \). Divide \( 10x^2 \) by \( x \), which gives \( 10x \).

   Multiply \( 10x \) by \( x + 1 \):
   \[
   10x \times (x + 1) = 10x^2 + 10x
   \]
   Subtract this from the current remainder:
   \[
   (10x^2 + 5x - 14) - (10x^2 + 10x) = -5x - 14
   \]

3. **Step 3: Divide the new leading term of the dividend by the leading term of the divisor.**

   The new leading term is \( -5x \). Divide \( -5x \) by \( x \), which gives \( -5 \).

   Multiply \( -5 \) by \( x + 1 \):
   \[
   -5 \times (x + 1) = -5x - 5
   \]
   Subtract this from the current remainder:
   \[
   (-5x - 14) - (-5x - 5) = -9
   \]

4. **Step 4: Express the result.**

   The quotient is \( 3x^2 + 10x - 5 \), and the remainder is \( -9 \). Therefore, the result of the division can be expressed as:
   \[
   3x^2 + 10x - 5 + \frac{-9}{x+1}
   \]
   or simply:
   \[
   3x^2 + 10x - 5 - \frac{9}{x+1}
   \]

This is the result of the long division.

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Here are 5 related questions for practice:
1. Divide \( 2x^3 + 5x^2 + 7x + 2 \) by \( x + 2 \).
2. Perform long division on \( x^3 - 6x^2 + 11x - 6 \) divided by \( x - 1 \).
3. Divide \( 4x^3 + 10x - 6 \) by \( x - 2 \).
4. Use long division to find the result of \( 5x^4 + 3x^2 + 7 \) divided by \( x + 1 \).
5. What is the remainder when \( x^3 + 8x^2 + 6x + 5 \) is divided by \( x - 3 \)?

**Tip:** When performing polynomial division, ensure that all terms are in descending order of degree, and include any missing terms with a coefficient of zero (e.g., \( 0x \)).

Use the long division method to find the result when 3x^3 + 13x^2 + 5x - 14 is divided by x + 1. If there is a remainder, express the result in the form q(x) + r(x) / b(x).

Learn how to perform polynomial long division with a step-by-step solution for dividing 3x^3 + 13x^2 + 5x - 14 by x + 1. This problem covers key algebraic concepts, such as polynomial division and expressing results with remainders. Suitable for students in Grades 10-12.

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