Question

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: 4, 6 - √7 Write the function in standard form. a) f (x) = x² - 10x + 23 b) f (x) = x² - 2x - 23 c) f (x) = x² - 10x - 23 d) f (x) = x² - 2x + 23

Answer 1

The polynomial function with the given zeros and requirements is f(x) = x² - (10 + √7)x + (3√7).

Step-by-step explanation:

To find a polynomial function with rational coefficients and a leading coefficient of 1 that has the given zeros, we need to use the zero product property. Since the zeros are 4 and 6 - √7, the corresponding factors are (x - 4) and (x - (6 - √7)). To find the polynomial function, we multiply these factors together and simplify:

f(x) = (x - 4)(x - (6 - √7))

f(x) = (x - 4)(x - 6 + √7)

f(x) = x² - 10x + 24 - √7x + 4√7 + √7 - 24

f(x) = x² - (10 + √7)x + (4√7 - √7)

f(x) = x² - (10 + √7)x + (3√7)

Therefore, the polynomial function in standard form is f(x) = x² - (10 + √7)x + (3√7).