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

[emily79]

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!

 

[Office_Shredder]

Definitely simplify it.

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

 

[tacman]

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!