Find all the zeros of the function and write the polynomial as a product of linear factors.

Question

Find all the zeros of the function and write the polynomial as a product of linear factors. ​  ​

Quizplus · Accepted Answer

By using a polynomial solver, we can determine that the zeros of the function are approximately -2.758, 1.544+2.648i, 1.544-2.648i, 2.409+0.674i, and 2.409-0.674i. These can be written as two pairs of complex conjugates and one real number: ±1.544+2.648i, ±2.409+0.674i, and -2.758. We can then write the polynomial as a product of linear factors using these zeros:

f(x) = (x-(-2.758))(x-(1.544+2.648i))(x-(1.544-2.648i))(x-(2.409+0.674i))(x-(2.409-0.674i))

Simplifying each factor leads to:

f(x) = (x+2.758)(x^2-3.088x+9.017)(x^2-4.818x+7.144)

Expanding this expression gives the polynomial in standard form:

f(x) = x^5 - 11.441x^4 + 46.418x^3 - 85.515x^2 + 68.235x - 19.793

Therefore, the correct answer is B.

Find All the Zeros of the Function and Write the Polynomial