Two polynomials P and D are given. Use either synthetic division or long division to divide 
P(x)
 by 
D(x),
 and express P in the form 
P(x) = D(x) · Q(x) + R(x).
P(x) = 5x3 − 8x2 − 5x,    D(x) = 5x − 8
P(x) = 
 

We are given the following polynomials:

\[
P(x) = 5x^3 - 8x^2 - 5x
\]
and
\[
D(x) = 5x - 8.
\]

We are tasked with dividing \( P(x) \) by \( D(x) \) and expressing it in the form:

\[
P(x) = D(x) \cdot Q(x) + R(x),
\]
where \( Q(x) \) is the quotient and \( R(x) \) is the remainder.

### Step 1: Set up the long division

We divide \( 5x^3 - 8x^2 - 5x \) by \( 5x - 8 \). Here's how long division works step by step:

#### 1. Divide the leading terms:
\[
\frac{5x^3}{5x} = x^2.
\]
This is the first term of \( Q(x) \).

#### 2. Multiply \( x^2 \) by \( 5x - 8 \):
\[
x^2 \cdot (5x - 8) = 5x^3 - 8x^2.
\]

#### 3. Subtract the result from \( P(x) \):
\[
(5x^3 - 8x^2 - 5x) - (5x^3 - 8x^2) = 0 - (-5x) = -5x.
\]

#### 4. Divide the next term:
\[
\frac{-5x}{5x} = -1.
\]
This is the second term of \( Q(x) \).

#### 5. Multiply \( -1 \) by \( 5x - 8 \):
\[
-1 \cdot (5x - 8) = -5x + 8.
\]

#### 6. Subtract again:
\[
(-5x) - (-5x + 8) = 0 - 8 = -8.
\]

### Step 2: Express the result

The quotient is \( Q(x) = x^2 - 1 \), and the remainder is \( R(x) = -8 \).

Thus, we can express \( P(x) \) as:

\[
P(x) = (5x - 8)(x^2 - 1) - 8.
\]

Would you like more details on the division process, or have any questions?

---

Here are 5 related questions to deepen understanding:

1. How would you apply synthetic division to this problem?
2. What is the degree of the remainder when dividing a cubic polynomial by a linear polynomial?
3. Can the remainder ever be greater than the divisor in polynomial division?
4. How can you verify the result of polynomial division?
5. What happens if the degree of \( D(x) \) is higher than the degree of \( P(x) \)?

**Tip:** Always double-check the subtraction steps in polynomial division to avoid small errors!

Two polynomials P and D are given. Use either synthetic division or long division to divide P(x) by D(x), and express P in the form P(x) = D(x) · Q(x) + R(x). P(x) = 5x^3 − 8x^2 − 5x, D(x) = 5x − 8.

Learn how to perform polynomial division of P(x) = 5x^3 - 8x^2 - 5x by D(x) = 5x - 8 using long division. This step-by-step guide breaks down the process and explains how to express the result in the form P(x) = D(x) · Q(x) + R(x). Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation