Question

Write a polynomial f(x) that satisfies the given conditions. Polynomial of lowest degree with zeros of -1/4 (multiplicity 2) and 2/3 (multiplicity 1) and with f(0)=2.

f(x)=?

Answer 1

Explanation:

To obtain a polynomial of lowest degree with the given zeros and leading coefficient, we can use the fact that if a polynomial has a root of multiplicity m, then the factor (x - r)^m appears in the polynomial. Therefore, we can write:

f(x) = A(x + 1/4)^2(x - 2/3)

where A is a constant that we need to determine to satisfy the condition f(0) = 2.

To find A, we substitute x = 0 into f(x) and set the result equal to 2:

f(0) = A(0 + 1/4)^2(0 - 2/3) = -A/27 = 2

Solving for A, we find that A = -54.

Therefore, the polynomial f(x) is:

f(x) = -54(x + 1/4)^2(x - 2/3)