Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Today, we are going to learn about the Remainder Theorem. This theorem tells us how to find the remainder when dividing a polynomial by a linear divisor.
What do you mean by a linear divisor?
Great question! A linear divisor is an expression that looks like x - a. For example, if we divide by x - 2, that's a linear divisor.
So, how do we actually find the remainder after division?
We simply evaluate the polynomial at the value of a. For instance, if we want to divide P(x) by x - 1, we evaluate P(1) to find the remainder.
Letβs apply the Remainder Theorem. Consider the polynomial P(x) = x^2 - 3x + 2. What do you think the remainder would be if we divide it by x - 1?
I think we need to calculate P(1)!
Exactly! Let's compute P(1). What do we get?
P(1) = 1^2 - 3(1) + 2 = 0. So, the remainder is 0!
Excellent! This means that x - 1 is a factor of the polynomial since the remainder is zero.
Now that we understand the Remainder Theorem, can anyone guess how it connects to the Factor Theorem?
If the remainder is zero, does that mean the divisor is a factor?
That's correct! If P(a) = 0, then x - a is a factor of P(x). This connection is very useful for factorization!
Can we use this theorem for higher-degree polynomials too?
Absolutely! The Remainder Theorem applies to any polynomial degree. It remains a powerful tool for analyzing polynomials.
When you divide with x minus a, the value of P at a, is the remainder, hooray!
Imagine you are baking a cake and you want to know how sweet it is (the remainder). You take a small piece (evaluate at a) to find out the sweetness left (the remainder) when you subtract what you've taken away.
Remainder after dividing, just plug in a and see what you are finding.
R.E.M: Remainder Equals Method - Simply evaluate at the point to find the remainder.
Term
Remainder Theorem
Click to flip
Term
Polynomial
Click to flip
An algebraic expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.
The amount left over after dividing a polynomial by a divisor.
An expression of the form x - a, where a is a constant.
A theorem that states if P(a) = 0 for a polynomial P(x), then x - a is a factor of P(x).
The Remainder Theorem states that the remainder of the division of a polynomial by a linear divisor can be determined by evaluating the polynomial at a specific value.
In this section, we explore the Remainder Theorem, explaining how to determine the remainder of a polynomial when divided by a linear polynomial. We illustrate this with examples, emphasizing the evaluation of the polynomial at a given point to find the remainder, showcasing the theorem's practical applications.
The Remainder Theorem is a vital concept in polynomial algebra that allows us to find the remainder of a polynomial when it is divided by a binomial of the form π₯ β π. If we have a polynomial function represented by π(π₯), the theorem states that the remainder of dividing π(π₯) by π₯ β π is equal to evaluating the polynomial at the point π, or mathematically written as:
Letβs consider a tangible example with the polynomial:
To find the remainder when divided by , we compute:
This implies that the remainder is 0, indicating that is a factor of the polynomial. This theorem paves the way for more advanced concepts like the Factor Theorem, allowing us to further manipulate and analyze polynomials effectively. Understanding the Remainder Theorem not only enhances polynomial division techniques but also has practical applications in determining polynomial roots and simplifying complex problems.