use synthetic division: 3x⁴+12x³-5x²-18x+8 ÷ x-4

To perform synthetic division, we'll divide the polynomial \( 3x^4 + 12x^3 - 5x^2 - 18x + 8 \) by \( x - 4 \). Here's a step-by-step guide:

### Step 1: Set up synthetic division

- The divisor is \( x - 4 \), so we use \( 4 \) as the number for synthetic division.
- The coefficients of the polynomial \( 3x^4 + 12x^3 - 5x^2 - 18x + 8 \) are:
  \[
  3, 12, -5, -18, 8
  \]

### Step 2: Perform synthetic division

1. Write the coefficients: \( 3, 12, -5, -18, 8 \).
2. Bring down the first coefficient, \( 3 \).
3. Multiply \( 3 \) by \( 4 \) (the divisor), and add to the next coefficient:
   \[
   4 \times 3 = 12, \quad 12 + 12 = 24
   \]
4. Multiply \( 24 \) by \( 4 \), and add to the next coefficient:
   \[
   4 \times 24 = 96, \quad 96 + (-5) = 91
   \]
5. Multiply \( 91 \) by \( 4 \), and add to the next coefficient:
   \[
   4 \times 91 = 364, \quad 364 + (-18) = 346
   \]
6. Multiply \( 346 \) by \( 4 \), and add to the next coefficient:
   \[
   4 \times 346 = 1384, \quad 1384 + 8 = 1392
   \]

### Step 3: Interpret the result

The result of synthetic division gives the quotient and remainder. The numbers from the synthetic division process are:
\[
3, 24, 91, 346 \quad \text{(Quotient)}
\]
and the last number \( 1392 \) is the remainder.

So, the quotient is:
\[
3x^3 + 24x^2 + 91x + 346
\]
and the remainder is \( 1392 \).

Thus, the result of the division is:
\[
3x^3 + 24x^2 + 91x + 346 + \frac{1392}{x - 4}
\]

### Do you have any questions or want more details?

Here are 5 related questions:
1. How do you check the result of synthetic division?
2. What is the difference between synthetic and long division in polynomials?
3. How can you solve synthetic division when dividing by a quadratic?
4. Can synthetic division be used for dividing by polynomials with higher degrees?
5. What is the significance of the remainder in synthetic division?

**Tip:** When using synthetic division, the divisor must be of the form \( x - c \).

Use synthetic division: 3x⁴+12x³-5x²-18x+8 ÷ x-4

Learn how to perform synthetic division on the polynomial 3x⁴ + 12x³ - 5x² - 18x + 8 divided by x - 4. This step-by-step solution covers key algebraic concepts such as synthetic division, finding the quotient and remainder, and interpreting the result. Suitable for students in Grades 9-12.

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