What is the coefficient of $x^3$ when $$x^4-3x^3 + 5x^2-6x + 1$$is multiplied by $$2x^3 - 3x^2 + 4x + 7$$and the like terms are combined?
I need to find the coefficient of $x^3$ in the product of the two polynomials. The last term, $-21x^3$, has degree $3$, so I will keep it. The second term, $18x^3$, has degree $3$, so I will keep it. The first term, $2x^3$, has degree $3$, so I will keep it. Now, I have four terms that have degree $3$: $-21x^3, 20x^3, 18x^3, 2x^3$. To find the coefficient of $x^3$ in the product, I need to add these four terms: $$-21x^3 + 20x^3 + 18x^3 +  {2x^3} = 19x^3$$ Therefore, the coefficient of $x^3$ in the product is 19. 