factor x^3-3x-52 — step-by-step solution

1) Let's factor in

ℝ

x^{3} - 3 x - 52

⇥1.1) Try the obvious solution first:

x = 4

⇥Let's evaluate the value of:

4^{3} - 3 \cdot 4 - 52

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

4^{3}

4 \cdot 4 \cdot 4

64

■Multiplications are done before additions and subtractions:

3 \cdot 4

12

⇥

64 - 12 - 52

⇥ⓘ Start from

64

and move back

12

on the number line:

64 - 12

52

⇥

52 - 52

⇥ⓘ Start from

52

and move back

52

on the number line, down to zero:

52 - 52

0

⇥

0

⇥Root found:

x = 4

⇥1.2) Compute the quotient and remainder of

(x^{3} - 3 x - 52)

(x - 4)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3} - 3 x - 52$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 4) = x^{3} - 4 x^{2}$	$(x^{3} - 3 x - 52) - (x^{3} - 4 x^{2}) = 4 x^{2} - 3 x - 52$
$4 x^{2} - 3 x - 52$	$\frac{4 x^{2}}{x}$	$4 x$	$4 x \cdot (x - 4) = 4 x^{2} - 16 x$	$(4 x^{2} - 3 x - 52) - (4 x^{2} - 16 x) = 13 x - 52$
$13 x - 52$	$\frac{13 x}{x}$	$13$	$13 \cdot (x - 4) = 13 x - 52$	$(13 x - 52) - (13 x - 52) = 0$

⇥Therefore, the result of the euclidean division is:

x^{2} + 4 x + 13

ⅰ

x^{3} - 3 x - 52

(x - 4) \cdot (x^{2} + 4 x + 13)

▶Factored expression:

(x - 4) \cdot (x^{2} + 4 x + 13)