### Instruction: Consider the polynomials  \[f(x)=4x^3+3x^2+2x+1\]and \[g(x)=3-4x+5x^2-6x^3.\]Find $c$ such that the polynomial $f(x)+cg(x)$ has degree 2.### Response: I need to find a constant $c$ that makes the degree of $f(x)+cg(x)$ equal to 2.
The degree of a polynomial is the highest power of $x$ that appears in it, so I want to eliminate the $x^3$ terms in $f(x)+cg(x)$.
To do that, I need to make the coefficient of $x^3$ in $f(x)+cg(x)$ equal to zero.
The coefficient of $x^3$ in $f(x)+cg(x)$ is the sum of the coefficients of $x^3$ in $f(x)$ and $cg(x)$.
The coefficient of $x^3$ in $f(x)$ is 4, and the coefficient of $x^3$ in $g(x)$ is -6, so the coefficient of $x^3$ in $cg(x)$ is -6c.
Therefore, the coefficient of $x^3$ in $f(x)+cg(x)$ is 4-6c.
I want this to be zero, so I need to solve the equation 4-6c=0.
Subtracting 4 from both sides, I get -6c=-4.
Dividing both sides by -6, I get c=4/6.
Simplifying the fraction, I get c=2/3.