Use synthetic division $$f ( x ) \text { by } x - k \text { for the given value of } k \text {. Then express } f ( x ) \text { in the form } f ( x ) = ( x - k ) q ( x ) + r$$ for the given value of k.
-$$f ( x ) = 2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 3 x ; \quad k = - 3$$
A) $f ( x ) = ( x - 3 ) \left( 2 x ^ { 3 } - 7 x ^ { 2 } + 6 x - 15 \right) + 45$
B) $f ( x ) = ( x + 3 ) \left( 2 x ^ { 3 } - 7 x ^ { 2 } + 6 x - 15 \right) + 45$
C) $f ( x ) = ( x + 3 ) \left( 2 x ^ { 3 } + 5 x ^ { 2 } + 3 \right) + 9$
D) $f ( x ) = ( x - 3 ) \left( 2 x ^ { 3 } - 5 x ^ { 2 } + 3 \right) + 9$

Question

Quizplus · Accepted Answer

Since $k = -3$, we use synthetic division with $-3$ to divide $2x^4 - x^3 - 15x^2 + 3x$. The correct division gives a quotient of $2x^3 - 7x^2 + 6x - 15$ and a remainder of $45$, fitting the form $f(x) = (x + 3)q(x) + r$.