Question

select the correct answer.
using synthetic division, find $(2x^4 - 3x^3 - 20x - 21) \div (x - 3)$.
\\(\bigcirc\\) a. \\(2x^3 + 3x^2 + 9x + 7\\)
\\(\bigcirc\\) b. \\(2x^4 + 3x^3 + 9x^2 + 7x\\)
\\(\bigcirc\\) c. \\(2x^4 + 3x^2 - 11x - 54\\)
\\(\bigcirc\\) d. \\(2x^3 + 3x^2 - 11x - 54\\)

Explanation:

Step1: Prepare for Synthetic Division

First, write the polynomial \(2x^4 - 3x^3 + 0x^2 - 20x - 21\) (we add \(0x^2\) to account for the missing \(x^2\) term) and the root of the divisor \(x - 3\), which is \(x = 3\).
Set up synthetic division with coefficients: \(2, -3, 0, -20, -21\) and root \(3\).

Step2: Perform Synthetic Division

Bring down the leading coefficient \(2\).
Multiply \(2\) by \(3\) to get \(6\), add to \(-3\): \(-3 + 6 = 3\).
Multiply \(3\) by \(3\) to get \(9\), add to \(0\): \(0 + 9 = 9\).
Multiply \(9\) by \(3\) to get \(27\), add to \(-20\): \(-20 + 27 = 7\).
Multiply \(7\) by \(3\) to get \(21\), add to \(-21\): \(-21 + 21 = 0\) (remainder is \(0\)).

The coefficients of the quotient polynomial are \(2, 3, 9, 7\) with degree one less than the original polynomial (since we divided by a linear term). So the quotient is \(2x^3 + 3x^2 + 9x + 7\).

Answer:

A. \(2x^3 + 3x^2 + 9x + 7\)