A polynomial P is given.P(x)=x^3+64 a)Find all zeros of P , real and complex

Trent Carpenter

Trent Carpenter

Answered question

2021-09-13

A polynomial P is given.
P(x)=x3+64
a)Find all zeros of P , real and complex.(Enter your answers as a comma-separated list. Enter your answers as a comma-separated list.)
x=?
b) Factor P completely.
P(z)=?

Answer & Explanation

Bella

Bella

Skilled2021-09-14Added 81 answers

Step 1
Given,
P(x)=x3+64
Step 2
We know that,
a3+b3=(a+b)(a2ab+b2)
Step 3
Part a
Finding the roots of P(x) , we get
P(x)=0
x3+64=0
x3+43=0
(x+4)(x24x+42)=0
(x+4)(x24x+16)=0
Either x+4=0  or  x24x+16=0
x=4  or  x=(4)±(4)24(1)(16)2(1)
x=4  or  x=4±16642
x=4  or  x=4±482
x=4  or  x=4±4i32
x=4  or  x=44i32,4+4i32
x=4  or  x=22i3,2+i23
  All zeros of P(x) are  x=4,22i3,2+i23
Step 4
Part b
P(x)=x3+43
=(x+4)(x24x+42)
=(x+4)(x(22i3))(x(2+2i3))
=(x+4)(x2+2i3)(x22i3)

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