Question

use synthetic division to find ( (4x^5 - 16x^4 + 28x^2 + 14x + 17) div (x - 2) ).
write your answer in the form ( q(x) + \frac{r}{d(x)} ), where ( q(x) ) is a polynomial, ( r ) is an integer, and ( d(x) ) is a linear polynomial. simplify any fractions.

Explanation:

Step1: Identify coefficients and root

The polynomial is \(4x^5 - 16x^4 + 0x^3 + 28x^2 + 14x + 17\) (note the missing \(x^3\) term with coefficient 0). The divisor is \(x - 2\), so the root \(c = 2\).

Step2: Set up synthetic division

Write the coefficients: \(4, -16, 0, 28, 14, 17\) and bring down the first coefficient (4).

Step3: Multiply and add

• Multiply \(4\times2 = 8\), add to \(-16\): \(-16 + 8 = -8\)
• Multiply \(-8\times2 = -16\), add to \(0\): \(0 + (-16) = -16\)
• Multiply \(-16\times2 = -32\), add to \(28\): \(28 + (-32) = -4\)
• Multiply \(-4\times2 = -8\), add to \(14\): \(14 + (-8) = 6\)
• Multiply \(6\times2 = 12\), add to \(17\): \(17 + 12 = 29\) (remainder \(r = 29\))

Step4: Form quotient polynomial

The quotient \(q(x)\) has coefficients \(4, -8, -16, -4, 6\), so \(q(x)=4x^4 - 8x^3 - 16x^2 - 4x + 6\) and \(d(x)=x - 2\)

Step5: Write the final form

The division result is \(q(x)+\frac{r}{d(x)} = 4x^4 - 8x^3 - 16x^2 - 4x + 6+\frac{29}{x - 2}\)

Answer:

\(4x^4 - 8x^3 - 16x^2 - 4x + 6+\frac{29}{x - 2}\)