Question

Write the left side of each equation as a product of linear factors, and state the solutions.

a. x^3 − 1 = 0
b. x^3 + 8 = 0
c. x^4 + 7x^2 + 10 = 0

Answer 1

a).
`x_(1)=1`,
`x_(2)=1`,
`x_(3)=1`

b).
`x_(1)=2`,
`x_(2)=2`,
`x_(3)=2`

c).
`x_(1) = √(-5)`,
`x_(2) = √(-5)`,
`x_(2) = √(-2)`,
`x_(4) = √(-2)`

Explanation:

a.

`x^(3)-1=0\\x^(3)=1\\ x=(1 )^{(1)/(3) }`

`x=1`

b.

`x^(3)+8=0\\x^(3)=-8\\ x=(-8 )^{(1)/(3) }`

`x= 2i`

c.

`x^(4) +7x^(2) +10=0\\x^(4)=u^(2) \\x^(2)=u\\ u^(2)+7u+10=0\\`

Using

`\frac{-b+/-\sqrt{7^(2)-4*a*c } }{2*a}`

`u=\frac{-7+/-\sqrt{7^(2)-4*10}}{2}=-(7)/(2) +/- (√(49-40) )/(2) \\ u=-(7)/(2) +/- (3)/(2) \\u_(1)= -5\\u_(2)= -2`

But u is no the factor is x so:

`x=√(u)\\ x_(1) =√(-5) =5i\\ x_(2) =√(-2) =2i`

Check:

`(√(5i))^(4) +7*(√(5i))^(2)  +10=0\\25*i^(4) +7*5*i^(2)+10=0\\ 25*1+7*5*-1+10\\25-35+10=0\\0=0`