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.

Guest Feb 2, 2021

#1**+1 **

A polynomial is of degree 3 if it is of the form \(h(x)=ax^3+bx^2+cx+d\).

\(f(x)+cg(x)=1-12x+3x^2-4x^3+2x^4+c(3-2x-6x^3+9x^4)\\ f(x)+cg(x)=1-12x+3x^2-4x^3+2x^4+3c-2cx-6cx^3+9cx^4\\ f(x)+cg(x)=(9c+2)x^4+(-6c-4)x^3+3x^2+(-2c-12)x+(3c+1)\)

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.

\(9c+2=0\\ c=-\frac{2}{9}\)

That's it!

Guest Feb 2, 2021

#1**+1 **

Best Answer

A polynomial is of degree 3 if it is of the form \(h(x)=ax^3+bx^2+cx+d\).

\(f(x)+cg(x)=1-12x+3x^2-4x^3+2x^4+c(3-2x-6x^3+9x^4)\\ f(x)+cg(x)=1-12x+3x^2-4x^3+2x^4+3c-2cx-6cx^3+9cx^4\\ f(x)+cg(x)=(9c+2)x^4+(-6c-4)x^3+3x^2+(-2c-12)x+(3c+1)\)

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.

\(9c+2=0\\ c=-\frac{2}{9}\)

That's it!

Guest Feb 2, 2021