Find ALL roots of the following.f(x)=8x^3+27

Question

x=

x=

a=

b=

c=

d=



Answer 1

x = cube root of -27/8 =- 3/2

there's 2 other imaginary roots found by factoring

there's a formula for the factors of the sum or difference of 2 cubes

(a+b)^2 = a^3 + 3a^2b + 3ab^2 + b^3

here a= 2x and b = 3

substitute and you get a quadratic for the 2nd factor. It has 2 imaginary roots

(2x+3)((2x)^3 + 3(2x)^2(3) +3(2x)(3)^2 + 3^3)

d= twice the coefficient of the x^2 term or 72

a= negative the coefficient of the x term or -54

b=i =sqr(-1)

c= -(a^2-4d(27)) = -54^2+4(27)72 =7776

the letters look a little strange as the quadratic formula is usually written

as x=-b/2a + or - (1/2a)sqr(b^2-4ac), same letters but different meaning

for ax^2 + bx + c

IF you don't know the formula, you do know or might guess a+b is a factor. Divide it into the cubic to get the 2nd factor

there are as many roots as the degree of the polynomial. cubic has 3 roots, although some may repeat if c=0 in this given problem, or in more traditional language and letters, if the discriminate = 0 = b^2-4ac. The "discriminate if >0 means all real roots, if <0 2 imaginary roots, or if= 0 then 2 repeating roots.