Form polynomial whose real zeros and degree are given. Zeros: 4,0, 9; degree: 3 Type a polynomial with integer coefficients and leading coefficient of 1. flx) = (Simplify your answer:
The Correct Answer and Explanation is:
To form a polynomial given the real zeros and the degree, you can use the factor theorem, which states that if rrr is a zero of a polynomial, then x−rx – rx−r is a factor of the polynomial.
Given:
- Zeros: 4, 0, and 9
- Degree: 3 (since there are 3 zeros)
We know that the polynomial will have the following factors:
- x−4x – 4x−4 for zero 4,
- x−0=xx – 0 = xx−0=x for zero 0,
- x−9x – 9x−9 for zero 9.
Thus, the polynomial can be written as:f(x)=(x−4)(x)(x−9)f(x) = (x – 4)(x)(x – 9)f(x)=(x−4)(x)(x−9)
Step-by-Step Expansion:
- First, expand (x−4)(x)(x – 4)(x)(x−4)(x):
(x−4)(x)=x2−4x(x – 4)(x) = x^2 – 4x(x−4)(x)=x2−4x
- Now, multiply the result by (x−9)(x – 9)(x−9):
(x2−4x)(x−9)(x^2 – 4x)(x – 9)(x2−4x)(x−9)
Use distributive property (also known as FOIL for binomials):x2(x−9)=x3−9x2x^2(x – 9) = x^3 – 9x^2×2(x−9)=x3−9×2−4x(x−9)=−4×2+36x-4x(x – 9) = -4x^2 + 36x−4x(x−9)=−4×2+36x
- Combine all terms:
x3−9×2−4×2+36xx^3 – 9x^2 – 4x^2 + 36xx3−9×2−4×2+36x=x3−13×2+36x= x^3 – 13x^2 + 36x=x3−13×2+36x
Thus, the polynomial is:f(x)=x3−13×2+36xf(x) = x^3 – 13x^2 + 36xf(x)=x3−13×2+36x
Explanation:
- The degree of the polynomial is 3, which matches the number of zeros.
- The leading coefficient is 1, as required (the coefficient of x3x^3×3 is 1).
- The polynomial has integer coefficients, and the factorization follows directly from the given zeros.
This polynomial satisfies the conditions: real zeros at 4, 0, and 9, and a degree of 3.
