Divide P(x) = 3x
2 −10x−4 by D(x) = x−3 and write in the form Q(x) + R(x)
D(x)
. Is D(x)
a factor of P(x)?


To divide \( P(x) = 3x^2 - 10x - 4 \) by \( D(x) = x - 3 \) and express it in the form \( Q(x) + \frac{R(x)}{D(x)} \), we can use polynomial long division or synthetic division. Let's proceed with the division.

### Step 1: Polynomial Long Division

We are dividing \( P(x) = 3x^2 - 10x - 4 \) by \( D(x) = x - 3 \).

1. Divide the first term of \( P(x) \) by the first term of \( D(x) \):
   \[
   \frac{3x^2}{x} = 3x
   \]
   So, the first term of \( Q(x) \) is \( 3x \).

2. Multiply \( 3x \) by \( x - 3 \) and subtract it from \( P(x) \):
   \[
   (3x^2 - 10x - 4) - (3x^2 - 9x) = -x - 4
   \]
   Now, the new polynomial to divide is \( -x - 4 \).

3. Divide \( -x \) by \( x \):
   \[
   \frac{-x}{x} = -1
   \]
   So, the next term of \( Q(x) \) is \( -1 \).

4. Multiply \( -1 \) by \( x - 3 \) and subtract:
   \[
   (-x - 4) - (-x + 3) = -7
   \]
   So, the remainder \( R(x) = -7 \).

### Step 2: Write the Result in the Desired Form

We have:
\[
Q(x) = 3x - 1 \quad \text{and} \quad R(x) = -7
\]

Thus, the division of \( P(x) \) by \( D(x) \) can be written as:
\[
P(x) = (3x - 1) + \frac{-7}{x - 3}
\]

### Step 3: Is \( D(x) = x - 3 \) a factor of \( P(x) \)?

For \( D(x) \) to be a factor of \( P(x) \), the remainder \( R(x) \) must be zero. Since the remainder is \( -7 \), \( D(x) = x - 3 \) is **not** a factor of \( P(x) \).

---

Would you like further details on any step or a more in-depth explanation of the process?

Here are 5 related questions to explore further:

1. How can we verify the remainder using synthetic division?
2. What happens if the remainder is zero in a polynomial division?
3. How do we factor \( P(x) \) completely?
4. What are the applications of polynomial division in algebraic simplifications?
5. Can we apply this method to divide polynomials of higher degrees?

**Tip:** Always check the degree of the remainder after polynomial division—it must be less than the degree of the divisor.

Divide P(x) = 3x^2 −10x−4 by D(x) = x−3 and write in the form Q(x) + R(x)/D(x). Is D(x) a factor of P(x)?

Learn how to divide P(x) = 3x^2 −10x−4 by D(x) = x−3 using polynomial long division and express it in the form Q(x) + R(x)/D(x). Discover how to determine if D(x) is a factor of P(x) using the remainder theorem. 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