factor x^3-8x^2+x+42 — step-by-step solution

1) Let's factor in

ℝ

x^{3}

- 8 x^{2}

+ x

+ 42

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

x = -2

⇥Let's compute the value of:

{(-2)}^{3}

- 8 \cdot {(-2)}^{2}

- 2

+ 42

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

{(-2)}^{3}

(-2) \cdot (-2) \cdot (-2)

-8

{(-2)}^{2}

(-2) \cdot (-2)

4

■Start from

-2

and move forward

42

on the number line, going past zero:

- 2 + 42

40

⇥

- 8 - 8 \cdot 4 + 40

⇥ⓘ Multiplications are done before additions and subtractions:

8 \cdot 4

32

⇥

- 8 - 32 + 40

⇥ⓘ Start from

-8

and move back

32

more on the number line, away from zero:

- 8 - 32

-40

⇥

- 40 + 40

⇥ⓘ Start from

-40

and move forward

40

on the number line, up to zero:

- 40 + 40

0

⇥

0

⇥Root found:

x = -2

⇥1.2) Perform polynomial long division of

(x^{3}

- 8 x^{2}

+ x

+ 42)

(x + 2)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- 8 x^{2}$ $+ x$ $+ 42$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x + 2) = x^{3} + 2 x^{2}$	$(x^{3}$ $- 8 x^{2}$ $+ x$ $+ 42) - (x^{3} + 2 x^{2}) = - 10 x^{2} + x + 42$
$- 10 x^{2} + x + 42$	$- \frac{10 x^{2}}{x}$	$- 10 x$	$- 10 x \cdot (x + 2) = - 10 x^{2} - 20 x$	$(- 10 x^{2} + x + 42) - (- 10 x^{2} - 20 x) = 21 x + 42$
$21 x + 42$	$\frac{21 x}{x}$	$21$	$21 \cdot (x + 2) = 21 x + 42$	$(21 x + 42) - (21 x + 42) = 0$

⇥Therefore, the result of the euclidean division is:

x^{2} - 10 x + 21

⇥1.3) Let's factor in

ℝ

(x^{2} - 10 x + 21)

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -10

and

c = 21

⇥

Δ = {(-10)}^{2} - 4 \cdot 1 \cdot 21

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

{(-10)}^{2}

(-10) \cdot (-10)

100

⇥

Δ = 100 - 4 \cdot 21

⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 21

4 \cdot (20 + 1)

4 \cdot 20 + 4

80 + 4

84

⇥

Δ = 100 - 84

⇥ⓘ Start from

100

and move back

84

on the number line:

100 - 84

16

⇥

Δ = 16

⇥Since

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-10) - \sqrt{16}}{2 \cdot 1}

⇥

x_{1} = \frac{10 - \sqrt{16}}{2}

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

\sqrt{16}

4

;

4^{2}

16

⇥

x_{1} = \frac{10 - 4}{2}

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

10 - 4

6

⇥

x_{1} = \frac{6}{2}

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

2

, their greatest common divisor:

\frac{6}{2}

3

⇥

x_{1} = 3

⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-10) + \sqrt{16}}{2 \cdot 1}

⇥

x_{2} = \frac{10 + \sqrt{16}}{2}

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

\sqrt{16}

4

;

4^{2}

16

⇥

x_{2} = \frac{10 + 4}{2}

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

10 + 4

14

⇥

x_{2} = \frac{14}{2}

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

2

, their greatest common divisor:

\frac{14}{2}

7

⇥

x_{2} = 7

▷The factor

(x^{2} - 10 x + 21)

factors as:

(x - 3) \cdot (x - 7)

▶Factored expression:

(x + 2) \cdot (x - 3) \cdot (x - 7)