Fill in the missing values below one at a time to find the quotient when 3, x, cubed, plus, 5, x, squared, minus, 14, x, minus, 163x
3
+5x
2
−14x−16 is divided by 3, x, plus, 83x+8.

To solve the problem of dividing the polynomial \(3x^3 + 5x^2 - 14x - 16\) by the binomial \(3x + 8\), we can use polynomial long division.

### Step 1: Divide the first term of the dividend by the first term of the divisor.
- The first term of the dividend is \(3x^3\).
- The first term of the divisor is \(3x\).

\[
\text{Quotient term} = \frac{3x^3}{3x} = x^2
\]

### Step 2: Multiply the entire divisor by this quotient term and subtract from the original polynomial.

\[
(x^2) \times (3x + 8) = 3x^3 + 8x^2
\]

Subtract:

\[
(3x^3 + 5x^2 - 14x - 16) - (3x^3 + 8x^2) = (5x^2 - 8x^2) - 14x - 16 = -3x^2 - 14x - 16
\]

### Step 3: Repeat the process with the new polynomial.

Now, divide the new first term \(-3x^2\) by the first term of the divisor \(3x\):

\[
\text{Quotient term} = \frac{-3x^2}{3x} = -x
\]

### Step 4: Multiply the divisor by \(-x\) and subtract.

\[
(-x) \times (3x + 8) = -3x^2 - 8x
\]

Subtract:

\[
(-3x^2 - 14x - 16) - (-3x^2 - 8x) = (-14x + 8x) - 16 = -6x - 16
\]

### Step 5: Divide the new first term \(-6x\) by \(3x\).

\[
\text{Quotient term} = \frac{-6x}{3x} = -2
\]

### Step 6: Multiply the divisor by \(-2\) and subtract.

\[
(-2) \times (3x + 8) = -6x - 16
\]

Subtract:

\[
(-6x - 16) - (-6x - 16) = 0
\]

The remainder is 0, which means the division is exact. The quotient is:

\[
\boxed{x^2 - x - 2}
\]

Would you like to go into more detail on any of these steps or have any other questions?

---

1. How do you find the quotient using synthetic division?
2. What happens if the remainder is not zero?
3. How would you verify the quotient obtained?
4. Can this method be used for dividing polynomials with higher degrees?
5. How does the divisor's form affect the complexity of division?

**Tip:** Always check the degrees of the terms during polynomial division to ensure correct alignment and subtraction.

Learn how to divide the polynomial 3x^3 + 5x^2 - 14x - 16 by the binomial 3x + 8 using polynomial long division. Detailed step-by-step solution included. Suitable for high school students studying algebra and polynomial division.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation