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

Question

this one really has me stumped

Answer 1

Hi Caden!

This polynomial will be degree 3 with the leading coefficient of 1. The three zeroes are 3, 4+i and 4-i bc if a+bi is a zero so is a-bi, aka the Conjugate postulate. With this info you can say the factors are (x-3)(x- (4+i) )(x-(4-i))

I would group it this way when beginning to multiply

(x-3)( (x-4) + i )( (x-4) -i )

I would first multiply the second two factors.

(x-4)² - (i)² [ this is a difference of squares (a+b)(a-b)

x² -4x -4x +16 - (-1)

x²-8x+16 +1

x² -8x +17

You can’t forget about (x-3)

(x-3)(x²-8x+17)

x³ -8x² +17x -3x² +24x -51

x³-11x²+ 41x -51

Thats the final answer. I hope this helped! Have a great day! Mrs. S

Answer 2

(x-3)(x-4-i)(x-4+i)

I will leave the multiplication to you.