Consider the polynomials:

F(x)= 4x^3 + 3x^2 + 2x +1
G(x)= 3 - 4x + 5x^2 - 6x^3
Find c such that the polynomial f(x)+cg(x) has degree 2.

well, the cube has to subtract out.

4-6c=0 solve for c

the answer is 2/3

To find the value of c such that the polynomial f(x) + cg(x) has degree 2, we need to understand how the degrees of polynomials behave when added.

The degree of a polynomial is determined by the term with the highest power of x. For polynomial f(x) = 4x^3 + 3x^2 + 2x + 1, the degree is 3 because the highest power of x is x^3. For polynomial g(x) = 3 - 4x + 5x^2 - 6x^3, the degree is also 3 because the highest power of x is also x^3.

Now, let's consider the polynomial f(x) + cg(x), where c is a constant factor. When we add two polynomials, the degrees of the terms are combined. In this case, we want the resulting polynomial to have a degree of 2. Therefore, the highest power of x has to be x^2.

To find the value of c, we need to determine what happens to the terms of f(x) and g(x) when they are added together. Since the degrees of the polynomial terms are combined, the term with the highest power of x should be the one that contributes to the degree 2.

In polynomial f(x), the highest power of x is x^3, so the term that contributes to the degree 2 is 0x^2. In polynomial g(x), the highest power of x is also x^3, so the term that contributes to the degree 2 is the coefficient of x^2, which is 5.

Therefore, for the polynomial f(x) + cg(x) to have a degree 2, the coefficient of x^2 in g(x) needs to match the coefficient of x^2 in f(x) + cg(x). In this case, we have 5 as the coefficient of x^2 in g(x) and 0 as the coefficient of x^2 in f(x) + cg(x).

Setting the coefficients equal to each other and solving for c, we have:
0 = 5c

Since 0 is equal to 5c, the value of c that satisfies this equation is 0.

Therefore, when c = 0, the polynomial f(x) + cg(x) has a degree 2.