Consider the polynomials \(f(x)=1-12x+3x^2-4x^3+5x^4\) and \(g(x)=3-2x-6x^3+9x^4.\) Find \(c \) such that the polynomial \(f(x)+cg(x)\) has degree 3.
You'll need to eliminate the x4 term; so you'll need a value for c, that when multiplied by 9x4 will cancel out the 5x4 term.
If you choose c to be -5/9, then
f(x) + c·g(x) = [ 1 - 12x + 3x2 -4x3 + 5x4 ] + (-5/9)·[ 3 - 2x - 6x3 + 9x4 ]
= [ 1 - 12x + 3x2 -4x3 + 5x4 ] + ·[ -5/3 + (10/9)x + (10/3)x3 - 5x4 ]
= -2/3 - 98/9x + 3x2 - (2/3)x3