factor x^4-6x^2-2x+4 — step-by-step solution

1) Let's factor in

ℝ

x^{4}

- 6 x^{2}

- 2 x

+ 4

⇥1.1) Try the obvious solution first:

x = -2

⇥Let's evaluate the value of:

{(-2)}^{4}

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

- 2 \cdot (-2)

+ 4

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

{(-2)}^{4}

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

16

{(-2)}^{2}

(-2) \cdot (-2)

4

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

- 2 \cdot (-2)

4

⇥

16

- 6 \cdot 4

+ 4

+ 4

⇥ⓘ Multiplications are done before additions and subtractions:

6 \cdot 4

24

⇥

16

- 24

+ 4

+ 4

⇥ⓘ Start from

16

and move back

24

on the number line, going below zero:

16 - 24

-8

⇥

- 8 + 4 + 4

⇥ⓘ Start from

-8

and move forward

4

on the number line, getting closer to zero:

- 8 + 4

-4

⇥

- 4 + 4

⇥ⓘ Start from

-4

and move forward

4

on the number line, up to zero:

- 4 + 4

0

⇥

0

⇥Root found:

x = -2

⇥1.2) Perform polynomial long division of

(x^{4}

- 6 x^{2}

- 2 x

+ 4)

(x + 2)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{4}$ $- 6 x^{2}$ $- 2 x$ $+ 4$	$\frac{x^{4}}{x}$	$x^{3}$	$x^{3} \cdot (x + 2) = x^{4} + 2 x^{3}$	$(x^{4}$ $- 6 x^{2}$ $- 2 x$ $+ 4) - (x^{4} + 2 x^{3}) = - 2 x^{3}$ $- 6 x^{2}$ $- 2 x$ $+ 4$
$- 2 x^{3}$ $- 6 x^{2}$ $- 2 x$ $+ 4$	$- \frac{2 x^{3}}{x}$	$- 2 x^{2}$	$- 2 x^{2} \cdot (x + 2) = - 2 x^{3} - 4 x^{2}$	$(- 2 x^{3}$ $- 6 x^{2}$ $- 2 x$ $+ 4) - (- 2 x^{3} - 4 x^{2}) = - 2 x^{2} - 2 x + 4$
$- 2 x^{2} - 2 x + 4$	$- \frac{2 x^{2}}{x}$	$- 2 x$	$- 2 x \cdot (x + 2) = - 2 x^{2} - 4 x$	$(- 2 x^{2} - 2 x + 4) - (- 2 x^{2} - 4 x) = 2 x + 4$
$2 x + 4$	$\frac{2 x}{x}$	$2$	$2 \cdot (x + 2) = 2 x + 4$	$(2 x + 4) - (2 x + 4) = 0$

⇥Therefore, the result of the euclidean division is:

x^{3}

- 2 x^{2}

- 2 x

+ 2

ⅰ

x^{4}

- 6 x^{2}

- 2 x

+ 4

(x + 2) \cdot (x^{3}

- 2 x^{2}

- 2 x

+ 2)

▶Factored expression:

(x + 2) \cdot (x^{3}

- 2 x^{2}

- 2 x

+ 2)