Solving Polynomials - Answers to Common Questions

[Kyoma]

i'm pretty new to polynomials and I have qns, which I do not know how to solve.

x⁵ + ax³ + bx² - 3 = (x² - 1)Q(x)- x - 2

Q(x) is a polynomial. Solve a and b.

I know the degree of Q(x) is 3, so I subst Q(x) into x³, but I got stuck.

Help. Thanks in advance.

 

[Mentallic]

You don't need to know what Q(x) is. The zeroes of $$x^2-1$$ are $$x=\pm 1$$ so if you substitute x=1 into the entire equality, that Q(x) part cancels since anything times zero is zero (your 'anything' is the Q(x)) and then you'll get an equation in a and b. Then substitute the other value of x to get another equation in a and b. Solve simultaneously and you're done!

 

[Kyoma]

Thank you very much.

How about this:

4x³ + 5x - 2 = (x+2)(x-1)P(x) + ax + b

Find a and b. Last qns, I seriously can't solve!

 

[Mentallic]

If you thought a bit more about what I posted you should have picked up on what to do. What values of x can you choose so that you eliminate the P(x)?