Thread:

1. Algebra polynomial

The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.

choices
f(x) = (x+4)(x^2+6x+10)
f(x) = (x-4)(x^2-6x-9
f(x) = (x-4)x^2-6x+10)
f(x) = (x-4)(x^2-6x+9)

2. Originally Posted by wvmcanelly@cableone.net

The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.
All such non-zero polynomials difference by a constant multiple. So we will find one which has 1 as its leading coefficient (called monic).

Now if f(x) is a polynomial in R[x] (meaning the real numbers) and if a+bi is a zero then a-bi is also a zero.
So if 3+i is a zero that means 3-i is a zero.

Thus,
$f(x) = (x-4)(x-(3+i))(x-(3-i))$
Multiply the last two factors together,
$f(x)=(x-4)(x^2 - 6x + 10)$

3. thanks, good not have figured that one out without you.
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