Consider the polynomials
f(x)=1-12x+3x^2-4x^3+2x^4.
and
g(x)=3-2x-6x^3+9x^4
Find c such that the polynomial f(x)+cg(x) has degree 3.
A polynomial is of degree 3 if it is of the form h(x)=ax3+bx2+cx+d.
f(x)+cg(x)=1−12x+3x2−4x3+2x4+c(3−2x−6x3+9x4)f(x)+cg(x)=1−12x+3x2−4x3+2x4+3c−2cx−6cx3+9cx4f(x)+cg(x)=(9c+2)x4+(−6c−4)x3+3x2+(−2c−12)x+(3c+1)
Expanding completely really was not necessary, but I did for completeness sake. In order to make this polynomial a degree 3 polynomial, the coefficient of x^4 should be 0.
9c+2=0c=−29
That's it!
A polynomial is of degree 3 if it is of the form h(x)=ax3+bx2+cx+d.
f(x)+cg(x)=1−12x+3x2−4x3+2x4+c(3−2x−6x3+9x4)f(x)+cg(x)=1−12x+3x2−4x3+2x4+3c−2cx−6cx3+9cx4f(x)+cg(x)=(9c+2)x4+(−6c−4)x3+3x2+(−2c−12)x+(3c+1)
Expanding completely really was not necessary, but I did for completeness sake. In order to make this polynomial a degree 3 polynomial, the coefficient of x^4 should be 0.
9c+2=0c=−29
That's it!
