Use the Remainder Theorem and synthetic division to find the function value. Verify your answers using another method.
$h ( x ) = x ^ { 3 } - 6 x ^ { 2 } - 5 x + 7 , \quad h ( - 8 )$
A) -849
B) -847
C) -851
D) -848
E) -845

Question

Use the Remainder Theorem and synthetic division to find the function value. Verify your answers using another method. 
$h ( x ) = x ^ { 3 } - 6 x ^ { 2 } - 5 x + 7 , \quad h ( - 8 )$
A) -849
B) -847
C) -851
D) -848
E) -845

Quizplus · Accepted Answer

To find $h(-8)$ using the Remainder Theorem, we perform synthetic division with -8. The Remainder Theorem states that if a polynomial $f(x)$ is divided by $x - k$, the remainder is $f(k)$. Here, $k = -8$.Synthetic division setup:$$\begin{array}{r|rrrr}-8 & 1 & -6 & -5 & 7 \\hline& & -8 & 112 & -856 \\end{array}$$Adding down the columns:$$\begin{array}{r|rrrr}-8 & 1 & -14 & 107 & -849 \\end{array}$$The remainder, which is the value of $h(-8)$, is -849.To verify using another method, we can directly substitute -8 into the polynomial:$$h(-8) = (-8)^3 - 6(-8)^2 - 5(-8) + 7 = -512 - 384 + 40 + 7 = -849.$$Both methods give the same result, confirming that the correct answer is -849.

Use the Remainder Theorem and Synthetic Division to Find the Function