PLEASE HELP FAST!!!!!

Question

write a polynomial function f of least degree that has rational coefficient a leading coefficients of 1 and the given zeros.

-2, 3 +√ 2

Answer 1

If a polynomial has rational coefficients and an irrational root, like 3+√2, then the conjugate, in this case 3-√2, must also be a root. The minimal polynomial for these two numbers will be a quadratic.

Putting x = 3+√2, we have x² = 11+6√2

Subtract 6x = 18 + 6√2

and add 7 to get 0.

So x²-6x+7 gives the roots 3+√2 and 3-√2.

But we need -2 as a root, too, so we need to multiply by x+2.

Answer 2

Since we want rational coefficients, 3 + √2 and 3 - √2 must both be roots..

By the Factor Theorem, if c is a root of a polynomial, then x - c is a factor of that polynomial.

A polynomial with the given roots is (x + 2)[x - (3 + √2)][x - (3 - √2)]

= (x + 2)[(x - 3) - √2][(x - 3) + √2] = (x + 2)[(x - 3)² - 2] = (x + 2)(x² - 6x + 7) = x³ - 4x² - 5x + 14