Question

use synthetic substitution to evaluate the given value and determine the remainder. f (x) = -5x⁵ - x⁴ + x³ - 6x² - 1 at x = -1

Explanation:

Step1: Identify coefficients

The polynomial \( f(x) = -5x^5 - x^4 + x^3 - 6x^2 - 1 \) can be written as \( f(x) = -5x^5 - x^4 + x^3 - 6x^2 + 0x - 1 \). So the coefficients are \(-5, -1, 1, -6, 0, -1\) (for \(x^5, x^4, x^3, x^2, x, \text{constant}\) respectively), and we use \(x = -1\) for synthetic substitution.

Step2: Set up synthetic division

Write the coefficients in a row: \(-5\), \(-1\), \(1\), \(-6\), \(0\), \(-1\), and the value \(x = -1\) to the left.

Step3: Perform synthetic substitution

• Bring down the first coefficient: \(-5\)
• Multiply by \(-1\): \(-5 \times (-1) = 5\), add to next coefficient: \(-1 + 5 = 4\)
• Multiply by \(-1\): \(4 \times (-1) = -4\), add to next coefficient: \(1 + (-4) = -3\)
• Multiply by \(-1\): \(-3 \times (-1) = 3\), add to next coefficient: \(-6 + 3 = -3\)
• Multiply by \(-1\): \(-3 \times (-1) = 3\), add to next coefficient: \(0 + 3 = 3\)
• Multiply by \(-1\): \(3 \times (-1) = -3\), add to last coefficient: \(-1 + (-3) = -4\)

Answer:

The remainder (which is \(f(-1)\)) is \(-4\)