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If a polynomial is divided by x - c, we can use the Remainder theorem to evaluate the polynomial at c.
The Remainder theorem:
If the polynomial f(x) is divided by x - c, then the remainder is f(c).
Example:
Given f(x) = x^3 - 4x^2 + 5x + 3, use the remainder theorem to find f(2).
Solution:
By the remainder theorem, if f(x) is divided by x - 2, then the remainder is f(2).
We can use the synthetic division to divide.
2] 1 -4 5 3
2 -4 2
__________
1 -2 1 5
The remainder is 5, so f(2) = 5
Check:
f(x) = x^3 - 4x^2 + 5x + 3
f(2) = (2)^3 - 4(2)^2 + 5(2) + 3 = 8 - 16 + 10 + 3 = 5
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What is polynomial division?
That means that you divide one polynomial by another polynomial. Basically, if you have polynomials "A" and "B", you look for a polynomial "C" and a remainder "R", such that: B x C + R = A ... such that the remainder has a lower degree than polynomial "B", the polynomial by which you are dividing. For example, if you divide by a polynomial of degree 3, the remainder must be of degree 2 or less.
If a polynomial is divided by (x-a) and the remainder equals zero then (x-a) is a factor of the polynomial?
false - apex
What is the polynomial factor theorem?
Suppose p(x) is a polynomial in x. Then p(a) = 0 if and only if (x-a) is a factor of p(x).
Is it false if a polynomial is divided by (x-a) and the remainder equals zero then (x-a) is a factor of the polynomial?
No, it’s true. It’s the same as saying if 60 is divided by 2 and the remainder equals zero (no remainder, so it divides perfectly), 2 is a factor of 60.
What is the remainder when 3 is synthetically divided into the polynomial -2x2 7x-9?
-6
