#1**+1 **

Let's first look at the possible combinations of terms that could give us an "x^{3}" term. There are 2 cases

1. Constant multiplied by an x^3 term(for example, x^{3 }* 6)

2. A first power of x multiplied by a second power

Let's start with the first case, with a constant multiplied by an x^{3 }term. We see that we are given two equations:

\(24x^4 + 6x^3 + 4x^2-7x - 5\) and \(6x^3 + 3x^2 + 3x + 4\). If we're looking at a constant multiplied by an x^{3 }term, there are two such "pairs" when we multiply these two polynomials:

\(6x^3*4 +6x^3*(-5) = 24x^3-30x^3 = -6x^3\)(Do you see how I got this? When you multiply the two polynomials, you can think of it as "cross multiplication" and take the desired pairs. That gives us our x^{3} term for our first case. Let's then move on to the second case:

A first power of x multiplied by a second power, or vice versa.

There are 2 such pairs that multiply together to form a third power of x here:

\(4x^2*3x +3x^2 *(-7x) = 12x^3 -21x^3 = -9x^3\)

Thus, we have our x^{3} coefficient for our first and second case, so all we have to do is now add them up.

\(-9x^3 +(-6x^3) = -15x^3\), giving us a coefficient of **-15**

jfan17 Mar 21, 2020