factor x^3+3x^2+3x+9 — step-by-step solution

1) Let's factorize in

ℝ

x^{3}

+ 3 x^{2}

+ 3 x

+ 9

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

x = -3

⇥Let's evaluate the value of:

{(-3)}^{3}

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

+ 3 \cdot (-3)

+ 9

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

{(-3)}^{3}

(-3) \cdot (-3) \cdot (-3)

-27

{(-3)}^{2}

(-3) \cdot (-3)

9

■Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

3 \cdot (-3)

-9

⇥

- 27

+ 3 \cdot 9

- 9

+ 9

⇥ⓘ Multiplications are done before additions and subtractions:

3 \cdot 9

27

■Start from

-9

and move forward

9

on the number line, up to zero:

- 9 + 9

0

⇥

- 27 + 27 + 0

⇥ⓘ Adding 0 changes nothing:

27 + 0

27

■Start from

-27

and move forward

27

on the number line, up to zero:

- 27 + 27

0

⇥

0

⇥Root found:

x = -3

⇥1.2) Compute the quotient and remainder of

(x^{3}

+ 3 x^{2}

+ 3 x

+ 9)

(x + 3)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $+ 3 x^{2}$ $+ 3 x$ $+ 9$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x + 3) = x^{3} + 3 x^{2}$	$(x^{3}$ $+ 3 x^{2}$ $+ 3 x$ $+ 9) - (x^{3} + 3 x^{2}) = 3 x + 9$
$3 x + 9$	$\frac{3 x}{x}$	$3$	$3 \cdot (x + 3) = 3 x + 9$	$(3 x + 9) - (3 x + 9) = 0$

⇥So the result of the euclidean division is:

x^{2} + 3

ⓘ A square being never negative,

x^{2} + 3

is never zero and cannot be factored in ℝ

ⅰ

x^{3}

+ 3 x^{2}

+ 3 x

+ 9

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

▶Factored expression:

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