intercepts of f(x)=x^3-x^2+x-1 — step-by-step solution

On the horizontal axis, y = 0: solve

x^{3}

- x^{2}

+ x

- 1 = 0

1) Let's solve in

ℝ

x^{3}

- x^{2}

+ x

- 1 = 0

⇥1.1) Let's start with the obvious candidate:

x = 1

⇥Let's evaluate the value of:

1^{3}

- 1^{2}

+ 1

- 1

⇥ⓘ Start from

1

and move back

1

on the number line, down to zero:

1 - 1

0

⇥

0 + 1 - 1

⇥ⓘ Adding 0 changes nothing:

0 + 1

1

■Start from

1

and move back

1

on the number line, down to zero:

1 - 1

0

⇥

0

⇥Solution found:

x = 1

⇥1.2) Perform polynomial long division of

(x^{3}

- x^{2}

+ x

- 1)

(x - 1)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- x^{2}$ $+ x$ $- 1$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 1) = x^{3} - x^{2}$	$(x^{3}$ $- x^{2}$ $+ x$ $- 1) - (x^{3} - x^{2}) = x - 1$
$x - 1$	$\frac{x}{x}$	$1$	$x - 1 = x - 1$	$(x - 1) - (x - 1) = 0$

⇥So the result of the euclidean division is:

x^{2} + 1

⇥1.3) Let's solve in

ℝ

x^{2} + 1 = 0

⇥

x^{2} = -1

⇥In ℝ, a square is never negative: no real number has

-1

as its square

⇥Solution:

x \in \emptyset

Solution obtained:

x = 1

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

⇥

0^{3}

- 0^{2}

+ 0

- 1

⇥ⓘ Adding 0 changes nothing:

0

+ 0

+ 0

- 1

-1

⇥

-1

▶Axis intercepts:

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