+0

# Hard Algebra Problem

0
197
1

Suppose that \$f\$ is a quadratic polynomial and \$g\$ is a cubic polynomial, and both \$f\$ and \$g\$ have a leading coefficient of \$1\$. What is the maximum degree of the polynomial \$(f(x))^3 - (g(x))^2 + f(x) - 1\$?

Anyone have an Idea on how to approach this problem?

Guest Jun 16, 2017
Sort:

#1
+5888
+1

f(x)  =  x2 + bx + c

g(x)  =  x3 + wx2 + yx + z

[f(x)]3 - [g(x)]2 + f(x) - 1

(x2+bx+c)3 - (x3+wx2+yx+z)2   +   x2+bx+c   -   1

(x2+bx+c)(x2​+bx+c)(x2​+bx+c)  -  (x3+wx2+yx+z)(x3+wx2+yx+z)   +   x2+bx+c   -   1

x4 + ...

We could continue, but by now I can see that the first term of   [f(x)]3   will be   1x6   and the first term of   [g(x)]2   will be   1x6   . Since   [g(x)]2    is being subtracted,  the   1x6   will cancel each other and go away. The next biggest exponent comes from  x3 * wx2  =  wx5 .

So  5   is the maximum degree.

hectictar  Jun 16, 2017

### 9 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details