Question

Use the remainder theorem to find P(-2) for P(x) = 2x^4 + 2x^3-x-5.Specifically, give the quotient and the remainder for the associated division and the value of P(-2).Quotient=?Remainder=?P(-2) = ?

Answer 1

The remainder theorem says that if a polynomial P(x) is divided by the binomial "x - a", then the remainder of the division is P(a). Therefore, to determine P(-2) we need to divide the polynomial P(x) by the binomial "x + 2". To do that we will use synthetic division. Using the coefficients of the polynomial.

`\begin{bmatrix}{2} & {2} & {0} \\ {\square} & {-4} & {\square} \\ {2} & {-2} & {\square}\end{bmatrix}\begin{bmatrix}{-1} & {-5} & {} \\ {\square} & {\square} & {} \\ {\square} & {\square} & {}\end{bmatrix}\begin{bmatrix}{-2} & {} & {} \\ {\square} & {} & {} \\ {\square} & {} & {}\end{bmatrix}`

Now we take the first coefficient and multiply that by -2 and add the result to the second coefficient

`\begin{bmatrix}{2} & {2} & {0} \\ {\square} & {-4} & {4} \\ {2} & {-2} & {4}\end{bmatrix}\begin{bmatrix}{-1} & {-5} & {} \\ {\square} & {\square} & {} \\ {\square} & {\square} & {}\end{bmatrix}\begin{bmatrix}{-2} & {} & {} \\ {\square} & {} & {} \\ {\square} & {} & {}\end{bmatrix}`

Now we multiply the result by -2 and add that to the next coefficient and so on until we get the last result which is the remainder, like this:

`\begin{bmatrix}{2} & {2} & {0} \\ {\square} & {-4} & {4} \\ {2} & {-2} & {4}\end{bmatrix}\begin{bmatrix}{-1} & {-5} & {} \\ {-8} & {18} & {} \\ {-9} & {13} & {}\end{bmatrix}\begin{bmatrix}{-2} & {} & {} \\ {\square} & {} & {} \\ {\square} & {} & {}\end{bmatrix}`

Therefore, the remainder is 13, therefore P(-2) = 13.

The quotient we find it by using the third row as coefficients of a polynomial of a grade one unit less than the original, that is:

`Q(x)=2x^3-2x^{2^{}}+4x-9`