Question

write a polynomial function ( f ) of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. write the polynomial in standard form.
3, 4 + i
( f(x) = square )

Explanation:

Step1: Identify conjugate zero

For rational coefficients, if $4+i$ is a zero, its conjugate $4-i$ is also a zero.

Step2: Write factor form

Use zeros to get factors: $(x-3)$, $(x-(4+i))$, $(x-(4-i))$
$f(x)=(x-3)(x-(4+i))(x-(4-i))$

Step3: Multiply complex factors

First simplify $(x-(4+i))(x-(4-i))$:
Let $a = x-4$, $b = i$, so $(a-b)(a+b)=a^2-b^2$
$(x-4)^2 - i^2 = x^2-8x+16 - (-1) = x^2-8x+17$

Step4: Multiply by linear factor

Multiply $(x-3)$ with $x^2-8x+17$:
$x(x^2-8x+17) - 3(x^2-8x+17)$
$=x^3-8x^2+17x-3x^2+24x-51$

Step5: Combine like terms

$x^3+(-8x^2-3x^2)+(17x+24x)-51$
$=x^3-11x^2+41x-51$

Answer:

$x^3-11x^2+41x-51$