Question

How do you use synthetic division and the Remainder Theorem to find the value of? f( -4) for the function `f(x) = x^5 + 8x^4 + 2x^3 - 6`?

Answer 1

`f(-4)=890`

Explanation:

The Remainder Theorem says
`color(white)("XX")f(a) = " remainder of "(f(x))/(x-a)`

So when `f(x)=x^5+8x^4+2x^3-6`
`color(white)("XX")f(-4) = " remainder of " (x^5+8x^4+2x^3-6)/(x+4)`

Synthetic Division provides a method for dividing polynomials which eliminates the need to write variables and generally reduces the number and complexity of calculations.
[For the simple case, like the example used here, the term "synthetic substitution" is commonly applied.]

Warning this next bit may sound complicated, but really isn't.
The "canonical coefficients" of a polynomial is the ordered set of coefficients of the terms of the polynomial beginning with the term with the greatest exponent and including every exponent variable term (including zero coefficient terms) down to and including the exponent of zero.

Using the given expression as an example:
`color(white)("XX")f(x)=x^5+8x^4+2x^3-6`
`color(white)("XXXX")=1*x^5+8*x^4+2*x^3+0*x^2+0*x^1-6*x^0`
and the canonical coefficients are `< 1, 8,2, 0, 0, -6 >`

To perform "synthetic division" (with a monic polynomial divisor) the quotient is replaced by the canonical coefficients of the numerator (in this case by `1 8 2 0 0 -6`)
and
the divisor is replaced by the negative of the canonical coefficients excluding the first coefficient (i.e. in this case by `(-4)`)

The process is carried out as illustrated below: