ANSWER
[tex]f( - 6) = 10[/tex]
EXPLANATION
The given function is
[tex]f(x) = {x}^{4} + 8 {x}^{3} + 10 {x}^{2} - 7x + 40[/tex]
According to the remainder theorem,
[tex]f( - 6) = R[/tex]
where R is the remainder when
[tex]f(x)[/tex]
is divided by
[tex]x + 6[/tex]
We therefore substitute the value of x and evaluate to obtain,
[tex]f( - 6) = { (- 6)}^{4} + 8 {( - 6)}^{3} + 10 { (- 6)}^{2} - 7( - 6)+ 40[/tex]
[tex]f( - 6) = 1296 + 8 ( - 216) + 10 (36) - 7( - 6)+ 40[/tex]
[tex]f( - 6) = 1296 - 1728 + 360 + 42+ 40[/tex]
This will evaluate to,
[tex]f( - 6) = 10[/tex]
Hence the value of the function when
[tex]x = - 6[/tex]
is
[tex]10[/tex]
According to the remainder theorem,
[tex]10[/tex]
is the remainder when
[tex]f(x) = {x}^{4} + 8 {x}^{3} + 10 {x}^{2} - 7x + 40[/tex]
is divided by
[tex]x + 6[/tex]
