Question

Use synthetic division to find the result when 2x^(4)+8x^(3)-13x^(2)-16x-5 is divided by x+5.

Answer 1

To divide the polynomial
`2x^4 + 8x^3 - 13x^2` - 16x - 5 by x + 5 using synthetic division, follow these steps: 1. Write the coefficients in descending order. 2. Perform the synthetic division. 3. The result is the quotient and the remainder.

Step-by-step explanation:

To divide the polynomial
`2x^4 + 8x^3 - 13x^2` - 16x - 5 by x + 5, we can use synthetic division. Here are the steps:

1. Write the coefficients of the polynomial in descending order. In this case, the coefficients are 2, 8, -13, -16, -5.

2. Place the constant term (-5) in the division box.

3. Bring down the first coefficient (2).

4. Multiply the number in the division box (5) by the coefficient you brought down (2) and write the result under the next coefficient (8).

5. Add the result to the next coefficient (-5 + 8 = 3) and write the sum under the next coefficient (-13).

6. Repeat steps 4 and 5 until you reach the last coefficient.

7. The last number in the division box is the remainder.

The result of dividing
`2x^4 + 8x^3 - 13x^2` - 16x - 5 by x + 5 is
`2x^3 + 3x^2` - 10x + 15 with a remainder of 70.

Answer 2

The result of dividing
`\(2x^4 + 8x^3 - 13x^2 - 16x - 5\)`by x + 5 using synthetic division is
`\(2x^3 - 2x^2 - 3x + 1\)`.

Step-by-step explanation:

Synthetic division is a method used to divide polynomials efficiently, especially when dividing by linear factors. In this case, we are dividing
`\(2x^4 + 8x^3 - 13x^2 - 16x - 5\) by \(x + 5\).`

The divisor x + 5 implies thatx = -5 is a root of the polynomial. We set up the synthetic division table using the coefficients of the polynomial and perform the division. The result in the last row of the synthetic division table corresponds to the coefficients of the quotient polynomial.

After completing the synthetic division, the quotient is
`\(2x^3 - 2x^2 - 3x + 1\)`. This means that
`\(2x^4 + 8x^3 - 13x^2 - 16x - 5\)` can be expressed as
`x + 5(2x^3 - 2x^2 - 3x + 1.`

Synthetic division is a useful technique for simplifying polynomial division, providing a straightforward way to find the quotient without performing long polynomial divisions.