Podcast
Questions and Answers
What is the Remainder Theorem used to do?
What is the Remainder Theorem used to do?
- Solve linear equations
- Evaluate polynomials at specific points and find the remainder when a polynomial is divided by a linear polynomial (correct)
- Graph polynomial functions
- Find the roots of a quadratic equation
If a polynomial f(x) is divided by (x - 3), what is the remainder according to the Remainder Theorem?
If a polynomial f(x) is divided by (x - 3), what is the remainder according to the Remainder Theorem?
- f(3) (correct)
- f(1)
- f(-3)
- f(0)
What is the remainder when f(x) = x^2 + 2x - 3 is divided by (x - 1)?
What is the remainder when f(x) = x^2 + 2x - 3 is divided by (x - 1)?
- 2
- 1 (correct)
- f(0)
- 0
What is the main idea behind the Remainder Theorem?
What is the main idea behind the Remainder Theorem?
When using the Remainder Theorem, what is being divided by what?
When using the Remainder Theorem, what is being divided by what?
What is NOT an application of the Remainder Theorem?
What is NOT an application of the Remainder Theorem?
Study Notes
Remainder Theorem
Definition
The Remainder Theorem is a powerful tool used to find the remainder when a polynomial is divided by a linear polynomial.
Statement
If a polynomial f(x) is divided by (x - a), then the remainder is f(a).
Explanation
- The theorem states that when a polynomial
f(x)is divided by(x - a), the remainder is the value of the polynomial atx = a. - In other words, the remainder is
f(a), which is the value of the polynomial whenxis equal toa. - This theorem can be used to evaluate polynomials at specific points without having to perform long division.
Example
- Find the remainder when
f(x) = x^3 - 2x^2 - 5x + 6is divided by(x - 2). - Using the Remainder Theorem, the remainder is
f(2) = 2^3 - 2(2)^2 - 5(2) + 6 = 4. - Therefore, the remainder is 4.
Applications
- The Remainder Theorem is used to:
- Evaluate polynomials at specific points.
- Find the remainder when a polynomial is divided by a linear polynomial.
- Factor polynomials.
- Solve polynomial equations.
Remainder Theorem
- The Remainder Theorem is a powerful tool used to find the remainder when a polynomial is divided by a linear polynomial.
Statement of the Theorem
- If a polynomial
f(x)is divided by(x - a), then the remainder isf(a).
Explanation and Key Points
- The theorem states that the remainder is the value of the polynomial at
x = a. - The remainder is
f(a), which is the value of the polynomial whenxis equal toa. - The theorem can be used to evaluate polynomials at specific points without performing long division.
Example Application
- To find the remainder when
f(x) = x^3 - 2x^2 - 5x + 6is divided by(x - 2), use the Remainder Theorem. - The remainder is
f(2) = 2^3 - 2(2)^2 - 5(2) + 6 = 4.
Applications of the Remainder Theorem
- Evaluate polynomials at specific points.
- Find the remainder when a polynomial is divided by a linear polynomial.
- Factor polynomials.
- Solve polynomial equations.
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Description
Learn about the Remainder Theorem, a powerful tool used to find the remainder when a polynomial is divided by a linear polynomial.
