Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: 2,3, square root of 5

Question

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: 2,3, square root of 5

Answer 1

Since 2,3, and √5 are roots of the polynomial function, the following are factors:

f(x) = (x - 2) (x - 3) (x - √5)

However if you multiply those factors together there will be some coefficients that are not rational.

So add the additional factor (x + √5)

f(x) = (x - 2) (x - 3)(x - √5)(x + √5)

f(x) = (x² - 5x + 6) (x² - 5) [Note the product of those last two factors results in a difference of squares, which is why (x + √5) was chosen].

Left to you to multiply the trinomial times the binomial.

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