zahid said:
1. For the polynomial, (a-4)x^7 + (2-3b)x^3 + (5c- 1), find the values of a, b and c if it is:
a.) Of degree 3
b.) Of degree 0
c.) Of degree 7 and monic
d.) The zero polynomial
I have no idea.
If it's as written, this question is all about making the coefficients zero...well, most of them.
a) Degree 3 means no terms higher than x^3 (that is, no x^4, x^5, x^6, x^7, etc.) So we want to kill the x^7 term -- the way to do that is to make the coefficient 0. So a-4=0 ==> a=4.
We want a polynomial of degree 3, so we must make sure that the coefficient of x^3 is not 0. So 2-3b != 0 ==> b != 2/3
c can be anything you like...it doesn't matter.
b) Similar idea -- kill both the x^7 and the x^3 terms. Note that there is often disagreement about what exactly constitutes a polynomial of degree zero. It definitely includes all polynomials which are just constants, but some regard 0 in that category, and some regard it as a class of its own. (I tend to regard it as a polynomial of degree zero, just like all the other constants.) If you include 0, then you don't care about c; if you don't include it, then you have to make c != 1/5. (Can you see why?)
c) Degree 7 -- you want the x^7 term in there. Monic -- look up your definitions again -- it means you want the leading coefficient (the coefficient of the highest order term) to be 1. So make the coefficient of x^7 equal to 1, and the others should be free variables.
d) The zero polynomial is exactly what it sounds like: 0 + 0x + 0x^2 + 0x^3 + 0x^4+... Kill all terms.
HTH.
