intercepts of f(x)=x^6-7x^3-8 — step-by-step solution

On the horizontal axis, y = 0: solve

x^{6} - 7 x^{3} - 8 = 0

⇥1) Let's solve in

ℝ

x^{6} - 7 x^{3} - 8 = 0

⇥Let's set:

X = x^{3}

⇥You can rewrite the expression as:

X^{2} - 7 X - 8 = 0

⇥Given the quadratic form

a X^{2} + b X + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -7

and

c = -8

⇥

Δ = {(-7)}^{2} - 4 \cdot 1 \cdot (-8)

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-7)}^{2}

(-7) \cdot (-7)

49

⇥

Δ = 49 - 4 \cdot (-8)

⇥ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-8)

32

⇥

Δ = 49 + 32

⇥ⓘ Start from

49

and move forward

32

on the number line:

49 + 32

81

⇥

Δ = 81

⇥Since

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

X_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-7) - \sqrt{81}}{2 \cdot 1}

⇥

X_{1} = \frac{7 - \sqrt{81}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{81}

9

;

9^{2}

81

⇥

X_{1} = \frac{7 - 9}{2}

⇥ⓘ The numerator and the denominator are computed before the division:

7 - 9

-2

⇥

X_{1} = - \frac{2}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{2}{2}

-1

⇥

X_{1} = -1

⇥The quadratic formula gives

X_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-7) + \sqrt{81}}{2 \cdot 1}

⇥

X_{2} = \frac{7 + \sqrt{81}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{81}

9

;

9^{2}

81

⇥

X_{2} = \frac{7 + 9}{2}

⇥ⓘ The numerator and the denominator are computed before the division:

7 + 9

16

⇥

X_{2} = \frac{16}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{16}{2}

8

⇥

X_{2} = 8

⇥Solutions found:

X = {-1, 8}

⇥2) Solving for x:

X = x^{3}

where

X = -1

⇥

x^{3} = -1

⇥Take the root:

- \sqrt[3]{1} = -1

. The power being odd,

x^{3} = -1

has a single real solution,

-1

⇥Solution found:

x = -1

⇥3) Let's solve for x:

X = x^{3}

using

X = 8

⇥

x^{3} = 8

⇥Take the root:

\sqrt[3]{8} = 2

. The power being odd,

x^{3} = 8

has a single real solution,

2

⇥Solution obtained:

x = 2

On the vertical axis, x = 0: calculate f(0)

⇥

0^{6} - 7 \cdot 0^{3} - 8

⇥ⓘ A product with a zero factor is zero:

-7 \cdot 0

0

■Adding 0 changes nothing:

0 - 8

-8

⇥

0 - 8

⇥ⓘ Adding 0 changes nothing:

0 - 8

-8

⇥

-8

▶Axis intercepts:

{(-1, 0), (2, 0), (0, -8)}