The Remainder Theorem is used to determine whether a linear binomial is a factor of a polynomial because it helps us factorize the polynomial more easily.
<h3>How to illustrate the theorem?</h3>
The remainder theorem states that if P(x) is a polynomial and x - a is a linear factor, the remainder when P(x) is divided by x - a is P(a). When P(a) = 0, then x - a is a factor of P(x).
Here, the remainder theorem is used to determine whether a linear binomial is a factor of a polynomial because it helps us factorize the polynomial more easily.
When the polynomial is divided by the linear factor, we obtain a polynomial of a lesser degree which can then be further factorized to obtain all the factors of our initial polynomial.
A linear binomial is the factor of a polynomial if the polynomial value is 0 at the zeros of the linear binomial
Let's assume a polynomial function is
P(x) = (x - 3)(x + 1)(x -2)
And a linear binomial is:
x - 3 = 0
We start by calculating the value of x in x - 3 = 0
l
x = 3
Next, we substitute x = 3 in P(x) = (x - 3)(x + 1)(x -2)
P(3) = (3 - 3)(3 + 1)(3 -2)
Evaluate
P(3) = 0
Since P(3) = 0, then the linear binomial x - 3 is a factor of P(x) = (x - 3)(x + 1)(x -2)
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This is still form ∞/∞ and we will use the derivative again:
=1/∞ = 0