The Factor Theorem and the Remainder Theorem work together to help find the zeros of a polynomial function.
The Factor Theorem: The Factor Theorem states that if a polynomial function f(x) has a factor (x - a), then f(a) = 0. In other words, if (x - a) is a factor of the polynomial, substituting a into the polynomial will result in a zero.
The Remainder Theorem: The Remainder Theorem states that if a polynomial function f(x) is divided by (x - a), then the remainder of that division is equal to f(a). In other words, if you divide the polynomial by (x - a), the remainder obtained will be the value of f(a).
By using these theorems together, we can find the zeros of a polynomial function. Here's an example:
Example:
Consider the polynomial function f(x) = x^3 - 4x^2 - 7x + 10. We want to find the zeros of this function.
Using the Factor Theorem:
To apply the Factor Theorem, we check if (x - a) is a factor of the polynomial. We can start by trying some values of a.
Let's try a = 1:
f(1) = (1)^3 - 4(1)^2 - 7(1) + 10 = 1 - 4 - 7 + 10 = 0
So, (x - 1) is a factor, and x = 1 is a zero of the function.
Using the Remainder Theorem:
To apply the Remainder Theorem, we can divide the polynomial f(x) by (x - a) and check the remainder. If the remainder is zero, then a is a zero of the function.
Let's try a = -2:
Dividing f(x) by (x - (-2)), we get:
f(x) = x^3 - 4x^2 - 7x + 10
Remainder = f(-2) = (-2)^3 - 4(-2)^2 - 7(-2) + 10 = -8 - 16 + 14 + 10 = 0
So, (x + 2) is a factor, and x = -2 is a zero of the function.
Therefore, the zeros of the function f(x) = x^3 - 4x^2 - 7x + 10 are x = 1 and x = -2.
Ways to determine if a number is a zero of a polynomial function:
1. By applying the Factor Theorem: If substituting the number into the polynomial gives a result of zero, then that number is a zero of the function.
2. By applying the Remainder Theorem: If dividing the polynomial by (x - a) gives a remainder of zero, then a is a zero of the function.
I hope this helps! :)