Write a polynomial in factored form with zeros 2,-2, and 1 . Each zero has multiplicity 1.
The correct answer and explanation is:
To construct a polynomial in factored form with given zeros 2, -2, and 1, each with multiplicity 1, follow these steps:
- Understand the concept of a zero and its multiplicity:
A zero of a polynomial is a value of xx where the polynomial equals zero. The multiplicity of a zero indicates how many times it appears as a factor in the polynomial. For example, a zero of 2 with multiplicity 1 means (x−2)(x – 2) appears once as a factor. - Write the factors for each zero:
If the zeros are 2, -2, and 1, the corresponding factors of the polynomial are:- For the zero 2, the factor is (x−2)(x – 2).
- For the zero -2, the factor is (x+2)(x + 2) because subtracting a negative number is equivalent to adding it.
- For the zero 1, the factor is (x−1)(x – 1).
- Multiply the factors together:
To create the polynomial, multiply all the factors: P(x)=(x−2)(x+2)(x−1)P(x) = (x – 2)(x + 2)(x – 1) - Expand the factored form:
First, expand the first two factors, (x−2)(x+2)(x – 2)(x + 2), which is a difference of squares: (x−2)(x+2)=x2−4(x – 2)(x + 2) = x^2 – 4 Now, multiply this result by the third factor, (x−1)(x – 1): (x2−4)(x−1)(x^2 – 4)(x – 1) Use distributive property (FOIL) to expand: =x2(x−1)−4(x−1)= x^2(x – 1) – 4(x – 1) =x3−x2−4x+4= x^3 – x^2 – 4x + 4
Thus, the polynomial is: P(x)=x3−x2−4x+4P(x) = x^3 – x^2 – 4x + 4
Explanation:
The polynomial P(x)=(x−2)(x+2)(x−1)P(x) = (x – 2)(x + 2)(x – 1) is constructed by considering the given zeros and their multiplicities. The factored form (x−2)(x+2)(x−1)(x – 2)(x + 2)(x – 1) directly corresponds to the zeros 2, -2, and 1. After expanding, we arrive at a cubic polynomial with integer coefficients. The degree of the polynomial is 3, which matches the number of distinct zeros. Each factor corresponds to a unique zero with multiplicity 1, and the expanded form shows the standard polynomial format.