Final answer:
To find the product of three binomials, (x - 2)(x + 4)(x - 7), we can use the distributive property. The result is x^3 - 5x^2 - 22x + 56. To find the three values of x that make the polynomial equal to zero, we solve the equation x^3 - 5x^2 - 22x + 56 = 0. This process involves techniques like factoring and graphing. The law that describes finding the roots of a polynomial equation is the Fundamental Theorem of Algebra.
Step-by-step explanation:
To find the product of (x - 2)(x + 4)(x - 7), we can use the distributive property multiple times.
First, we distribute the first two binomials:
(x - 2)(x + 4) = x(x) + x(4) - 2(x) - 2(4) = x^2 + 4x - 2x - 8 = x^2 + 2x - 8
Next, we distribute the result above with the third binomial:
(x^2 + 2x - 8)(x - 7) = x(x^2) + x(2x) - x(8) - 7(x^2) - 7(2x) + 7(8) = x^3 + 2x^2 - 8x - 7x^2 - 14x + 56 = x^3 - 5x^2 - 22x + 56
Therefore, the product of (x - 2)(x + 4)(x - 7) is x^3 - 5x^2 - 22x + 56.
To find the three values of x that make the cubic polynomial equal to zero, we set the polynomial equal to zero and solve for x:
x^3 - 5x^2 - 22x + 56 = 0
This equation can be solved using various methods such as factoring, graphing, or the Rational Root Theorem.
The famous law that describes finding the roots of a polynomial equation is known as the Fundamental Theorem of Algebra.