Pre-Calculus Homework help.?

Things you need to know

1) If 'a' is a zero, then (x-a) is a factor

2) Complex roots always occur in conjugate pairs, so if (a+i) is a factor, then so is (a-i), where i is the imaginary number

3) i² = -1

4) Expand

(x-(4+i)) (x-(4-i)) (x-6)

= x³ -14x² +65x - 102

5) Expand

(x-1)(x-2) (x-(2+i)) (x-(2-i))

= x⁴ - 7x³ + 19x² - 23x + 10

Given a complex polynomial function whose coefficients are real numbers,

a) find the remaining zeros of f, then

b) find a polynomial, in expanded form, w/real coefficients that has zeros

4) degree 3; zeros: 4+i, 6

Since 4 + i is a zero, 4 – i is also a zero

Quadratic equation with 4 + i and 4 – i as zeroes:

x² – ((4 + i) + (4 – i))x + (4 + i)(4 – i)

x² – 8x + 17

Other factor is x – 6

f(x) = (x – 6)(x² – 8x + 17)

..... = x(x² – 8x + 17) – 6(x² – 8x + 17)

..... = x³ – 8x² + 17x – 6x² + 48x – 102

..... = x³ – 14x² + 65x – 102

5) Degree 4; zeros: 1, 2, 2+i

Since 2 + i is a zero, 2 – i is also a zero

Quadratic equation with 2 + i and 2 – i as zeroes:

x² – ((2 + i) + (2 – i))x + (2 + i)(2 – i)

x² – 4x + 5

Quadratic equation with 1 and 2 as zeroes

x² – (1 + 2)x + (1)(2)

x² – 3x + 2

f(x) = (x² – 4x + 5)(x² – 3x + 2)

..... = x²(x² – 3x + 2) – 4x(x² – 3x + 2) + 5(x² – 3x + 2)

..... = x⁴ – 3x³ + 2x² – 4x³ + 12x² – 8x + 5x² – 15x + 10

..... = x⁴ – 3x³ – 4x³ + 2x² + 12x² + 5x² – 8x – 15x + 10

..... = x⁴ – 7x³ + 19x² – 23x + 10

V = (s)(s)(h) = one hundred SA = (s)(s) + 4(s)(h) = one hundred twenty from the 1st equation you get h = one hundred/(s*s) plug that into the 2d equation (s)(s) + 4(one hundred)/s = one hundred twenty multiply each and every thing with the aid of s and you get s^3 + 4 hundred = 120s you will have a graphing calculator, with the aid of which you will discover the dimensions of the area after which the top of the field to discover the optimum quantity V = (s)(s)(h) = max SA = (s)(s) + 4(s)(h) = one hundred twenty from the 2d equation you get h = (one hundred twenty - s^2) / (4s) plug it into the 1st equation V = (s^2)(one hundred twenty - s^2) / (4s) simplify with the aid of doing away with an s V = 30s- (s^3)/4 so plug that into your graphing calculator and get your optimum