Solve the Remainder Theorem with x^2-4x^2+3

In summary, the Remainder Theorem states that when a polynomial is divided by (x-a), the remainder is equal to f(a). To solve the theorem, the polynomial must be set up as f(x) and the value of (x-a) must be plugged in to find the remainder. The Remainder Theorem can be used for all polynomials, but only when the divisor is of the form (x-a). Its purpose is to find the remainder of a polynomial division, which can be useful in solving equations and finding roots. However, it has limitations as it only applies to (x-a) division and does not provide the full solution.
  • #1
emily79
1
0
remainder theorem...?

Find the value of 'a' and 'b' and the remaining factor if the expression ax^3-11x^2+bx+3 is divisible by x^2-4x^2+3


do i simplify x^2-4x^2+3 and then substitute for x?

im so lostt!
 
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  • #2
Definitely simplify it.

A good place to start: If one polynomial is divisible by another one, what does that say about their roots?
 
  • #3


To solve the remainder theorem, we first need to simplify the polynomial x^2-4x^2+3. This can be done by combining like terms, which gives us -3x^2+3. Now, we can substitute this expression for x in the given polynomial ax^3-11x^2+bx+3.

This gives us the following equation: a(-3x^2+3)^3-11(-3x^2+3)^2+b(-3x^2+3)+3. Since we know that this expression is divisible by x^2-4x^2+3, we can set up a system of equations to solve for the values of 'a' and 'b'.

The first equation will be the remainder theorem, which states that when a polynomial is divided by a linear factor, the remainder is equal to the value of the polynomial when the factor is substituted in. In this case, it would be: a(-3x^2+3)^3-11(-3x^2+3)^2+b(-3x^2+3)+3 = 0.

The second equation will come from the fact that the polynomial is divisible by x^2-4x^2+3, which means that when we divide the polynomial by this factor, the quotient should be a polynomial with no remainder. This gives us the equation: (ax^3-11x^2+bx+3)/(x^2-4x^2+3) = ax+b.

Now, we can solve this system of equations for 'a' and 'b'. Once we have the values for 'a' and 'b', we can find the remaining factor by dividing the original polynomial by (ax+b). This will give us the quotient, which is the remaining factor.

I hope this helps clarify the process of solving the remainder theorem with x^2-4x^2+3 as the divisor. Remember to always simplify the divisor and substitute it for x before setting up the system of equations. Good luck!
 

1. What is the Remainder Theorem?

The Remainder Theorem states that when a polynomial f(x) is divided by (x-a), the remainder is equal to f(a).

2. How do I solve the Remainder Theorem with x^2-4x^2+3?

To solve the Remainder Theorem with x^2-4x^2+3, you would first need to set up the polynomial as f(x) = x^2-4x^2+3. Then, you would plug in the value of (x-a) and solve for the remainder using f(a).

3. Can the Remainder Theorem be used for all polynomials?

Yes, the Remainder Theorem can be used for all polynomials, as long as the divisor is of the form (x-a).

4. What is the purpose of using the Remainder Theorem?

The Remainder Theorem is used to find the remainder of a polynomial division, which can be helpful in solving more complex equations or finding the roots of a polynomial.

5. Are there any limitations to the Remainder Theorem?

The Remainder Theorem only applies to polynomial division by (x-a), and cannot be used for other types of division. Additionally, it only gives the remainder and does not provide the complete solution to the division problem.

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