How Does the Remainder Theorem Simplify Polynomial Division?

In summary, the Remainder Theorem is a mathematical theorem used to find the remainder when dividing a polynomial by a linear function. It can also be used to evaluate polynomial functions and find factors of polynomials. It is often confused with the Factor Theorem, which is used to determine roots and factors of polynomials. Some common mistakes when using the Remainder Theorem include forgetting to include the negative sign and using the wrong value of x in the linear function. However, it can be applied to any type of polynomial as long as the divisor is a linear function.
  • #1
Jordan1994
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0
Q2.) Show all working out.

a) Find the remainder when \(\displaystyle x^3+2x^2-5x-3\) is divided by \(\displaystyle x-2\).

b) Find the remainder when \(\displaystyle x^3-3x^2-x+3\) is divided by \(\displaystyle x-3\).
 
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  • #2
Jordan1994 said:
Q2.) Show all working out.

a) Find the remainder when \(\displaystyle x^3+2x^2-5x-3\) is divided by \(\displaystyle x-2\).

b) Find the remainder when \(\displaystyle x^3-3x^2-x+3\) is divided by \(\displaystyle x-3\).
You titled this "remainder theorem question". What does the "remainder theorem" say?
 

FAQ: How Does the Remainder Theorem Simplify Polynomial Division?

What is the Remainder Theorem?

The Remainder Theorem is a mathematical theorem that states that when a polynomial function is divided by a linear function of the form x-a, the remainder is equal to the value of the polynomial at x=a.

How do you use the Remainder Theorem to solve problems?

To use the Remainder Theorem, you first need to divide the polynomial by the linear function using long division or synthetic division. The remainder will then be equal to the value of the polynomial at the value of x in the linear function. This can be used to find the remainder when dividing polynomials, evaluate polynomial functions at a specific value, or find factors of polynomials.

What is the difference between the Remainder Theorem and the Factor Theorem?

The Remainder Theorem and the Factor Theorem are closely related, but they are used for different purposes. The Remainder Theorem is used to find the remainder when dividing polynomials, while the Factor Theorem is used to determine whether a value is a root or factor of a polynomial.

What are some common mistakes when using the Remainder Theorem?

One common mistake when using the Remainder Theorem is forgetting to include the negative sign in the linear function. Another mistake is using the wrong value of x in the linear function, which can result in an incorrect remainder. It's important to double check your calculations and make sure all signs and values are correct.

Can the Remainder Theorem be applied to any type of polynomial?

Yes, the Remainder Theorem can be applied to any type of polynomial, including quadratic, cubic, and higher degree polynomials. However, the divisor must be a linear function of the form x-a. If the divisor is not linear, then the Remainder Theorem cannot be used.

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