2. The Remainder Theorem and the Factor Theorem
This section discusses the historical method of solving higher degree polynomial equations.
As we discussed in the previous section Polynomial Functions and Equations, a polynomial function is of the form:
f(x) = a0xn + a1xn-1 + a2xn-2 + ... + an
a0 ≠ 0 and
n is a positive integer, called the degree of the polynomial.
f(x) = 7x5 + 4x3 − 2x2 − 8x + 1 is a polynomial function of degree 5.
First, let's consider what happens when we divide numbers.
Example (a): Say we try to divide `13` by `5`. We will get the answer `2` and have a remainder of `3`. We could write this as:
`13/5 = 2 + 3/5`
Another way of thinking about this example is:
`13 = 2 × 5 + 3`
Example (b), Long Division: In primary school, you may have learned to divide larger numbers as follows. Let's divide `3,756` by `23`.
|`23`||We multiply `23` by `1 = 23`.|
|`145`||`37-23 = 14`. Then bring down the `5`.|
|`138`||Multiply `23` by `6=138`.|
|`76`||`145-138=7`. Bring down the `6`.|
|`69`||Multiply `23` by `3=69`.|
|`7`||`76-69=7`. This is the remainder.|
So we can conclude `3,756 -: 23 = 163 + 7/23`, or putting it another way, `3,756 = 163xx23 + 7`.
Division of polynomials is an extension of our number examples.
If we divide a polynomial by (x − r), we obtain a result of the form:
f(x) = (x − r) q(x) + R
where q(x) is the quotient and R is the remainder.
Let's now see an example of polynomial division.
Divide f(x) = 3x2 + 5x − 8 by (x − 2).
The Remainder Theorem
Consider f(x) = (x − r)q(x) + R
Note that if we let x = r, the expression becomes
f(r) = (r − r) q(r) + R
f(r) = R
This leads us to the Remainder Theorem which states:
If a polynomial f(x) is divided by (x − r) and a remainder R is obtained, then f(r) = R.
Use the remainder theorem to find the remainder for Example 1 above, which was divide f(x) = 3x2 + 5x − 8 by (x − 2).
By using the remainder theorem, determine the remainder when
3x3 − x2 − 20x + 5
is divided by (x + 4).
The Factor Theorem
The Factor Theorem states:
If the remainder f(r) = R = 0, then (x − r) is a factor of f(x).
The Factor Theorem is powerful because it can be used to find roots of polynomial equations.
Is (x + 1) a factor of f(x) = x3 + 2x2 − 5x − 6?
1. Find the remainder R by long division and by the Remainder Theorem.
(2x4 - 10x2 + 30x - 60) ÷ (x + 4)
2. Find the remainder using the Remainder Theorem
(x4 − 5x3 + x2 − 2x + 6) ÷ (x + 4)
3. Use the Factor Theorem to decide if (x − 2) is a factor of
f(x) = x5 − 2x4 + 3x3 − 6x2 − 4x + 8.
4. Determine whether `-3/2` is a zero (root) of the function:
f(x) = 2x3 + 3x2 − 8x − 12.
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