Introduction:

To find a zero of the polynomial p(x) = 2x^3, we need to solve for x when p(x) equals zero. In other words, we are looking for the value(s) of x that make the polynomial equal to zero.

Zero of a Polynomial:

A zero of a polynomial is a value of x that makes the polynomial equal to zero. It is also known as a root or a solution of the polynomial equation. In this case, we need to find the value(s) of x for which p(x) = 0.

Finding the Zero:

To find the zero of the polynomial p(x) = 2x^3, we set p(x) equal to zero and solve for x:

2x^3 = 0

Zero Product Property:

The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In this case, we have the factor 2x^3.

Setting Factors to Zero:

To apply the zero product property, we set each factor equal to zero and solve for x:

2x^3 = 0

Setting the factor 2x^3 equal to zero, we get:

2x^3 = 0
x^3 = 0

Finding the Cube Root:

To solve for x, we need to find the cube root of 0. The cube root of a number is the number that, when multiplied by itself three times, gives the original number. In this case, the cube root of 0 is 0.

Therefore, the zero of the polynomial p(x) = 2x^3 is x = 0.

Summary:

The zero of the polynomial p(x) = 2x^3 is x = 0. This means that when x is equal to 0, the polynomial p(x) equals zero. The zero product property is used to set each factor of the polynomial equal to zero and solve for x. In this case, the factor 2x^3 is set equal to zero, and the cube root of 0 is found to be 0. Thus, x = 0 is the zero of the polynomial.

P(x)=02x-3=02x=3x=3/2