factor x^4+2x^3+1 — step-by-step solution

⚠ 2x is read as the product 2*x: for a power, write 2^x

Let's factorize in

ℝ

x^{4} + 2 x^{3} + 1

Try the obvious solution first:

x = -1

Let's compute the value of:

{(-1)}^{4} + 2 \cdot {(-1)}^{3} + 1

ℹ

{(-1)}^{4}

1

‖

{(-1)}^{3}

-1

‖

2 \cdot (-1)

-2

1 - 2 + 1

- 1 + 1

0

Factor found:

x = -1

Compute the quotient and remainder of

x^{4} + 2 x^{3} + 1

x + 1

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{4} + 2 x^{3} + 1$	$\frac{x^{4}}{x}$	$x^{3}$	$x^{3} \cdot (x + 1) = x^{4} + x^{3}$	$(x^{4} + 2 x^{3} + 1) - (x^{4} + x^{3}) = x^{3} + 1$
$x^{3} + 1$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x + 1) = x^{3} + x^{2}$	$(x^{3} + 1) - (x^{3} + x^{2}) = - x^{2} + 1$
$- x^{2} + 1$	$- \frac{x^{2}}{x}$	$- x$	$- x \cdot (x + 1) = - x^{2} - x$	$(- x^{2} + 1) - (- x^{2} - x) = x + 1$
$x + 1$	$\frac{x}{x}$	$1$	$x + 1 = x + 1$	$(x + 1) - (x + 1) = 0$

So the result of the euclidean division is:

x^{3}

+ x^{2}

- x

+ 1

▶Factored expression:

(x + 1) \cdot (x^{3}

+ x^{2}

- x

+ 1)