factorize x^4-4x^3-7x^2+22x+24 — step-by-step solution

1) Let's factor in

ℝ

x^{4}

- 4 x^{3}

- 7 x^{2}

+ 22 x

+ 24

Let's start with the obvious candidate:

x = -1

2) Let's evaluate the value of:

{(-1)}^{4}

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

- 7 \cdot {(-1)}^{2}

+ 22 \cdot -1

+ 24

1

- 4 \cdot -1

- 7 \cdot 1

- 22

+ 24

- 21

+ 4

- 7

+ 24

- 17 - 7 + 24

- 24 + 24

0

Factor found:

x = -1

3) Compute the quotient and remainder of

x^{4}

- 4 x^{3}

- 7 x^{2}

+ 22 x

+ 24

x + 1

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{4}$ $- 4 x^{3}$ $- 7 x^{2}$ $+ 22 x$ $+ 24$	$\frac{x^{4}}{x}$	$x^{3}$	$x^{3} \cdot (x + 1) = x^{4} + x^{3}$	$(x^{4}$ $- 4 x^{3}$ $- 7 x^{2}$ $+ 22 x$ $+ 24) - (x^{4} + x^{3}) = - 5 x^{3}$ $- 7 x^{2}$ $+ 22 x$ $+ 24$
$- 5 x^{3}$ $- 7 x^{2}$ $+ 22 x$ $+ 24$	$- \frac{5 x^{3}}{x}$	$- 5 x^{2}$	$- 5 x^{2} \cdot (x + 1) = - 5 x^{3} - 5 x^{2}$	$(- 5 x^{3}$ $- 7 x^{2}$ $+ 22 x$ $+ 24) - (- 5 x^{3} - 5 x^{2}) = - 2 x^{2} + 22 x + 24$
$- 2 x^{2} + 22 x + 24$	$- \frac{2 x^{2}}{x}$	$- 2 x$	$- 2 x \cdot (x + 1) = - 2 x^{2} - 2 x$	$(- 2 x^{2} + 22 x + 24) - (- 2 x^{2} - 2 x) = 24 x + 24$
$24 x + 24$	$\frac{24 x}{x}$	$24$	$24 \cdot (x + 1) = 24 x + 24$	$(24 x + 24) - (24 x + 24) = 0$

So the result of the euclidean division is:

x^{3}

- 5 x^{2}

- 2 x

+ 24

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

- 5 x^{2}

- 2 x

+ 24)

4) Let's factorize in

ℝ

(x^{3}

- 5 x^{2}

- 2 x

+ 24)

Try the obvious solution first:

x = -2

5) Let's compute the value of:

{(-2)}^{3}

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

- 2 \cdot -2

+ 24

- 8

- 5 \cdot 4

+ 4

+ 24

- 4 - 20 + 24

- 24 + 24

0

Factor identified:

x = -2

6) Compute the quotient and remainder of

x^{3}

- 5 x^{2}

- 2 x

+ 24

x + 2

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- 5 x^{2}$ $- 2 x$ $+ 24$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x + 2) = x^{3} + 2 x^{2}$	$(x^{3}$ $- 5 x^{2}$ $- 2 x$ $+ 24) - (x^{3} + 2 x^{2}) = - 7 x^{2} - 2 x + 24$
$- 7 x^{2} - 2 x + 24$	$- \frac{7 x^{2}}{x}$	$- 7 x$	$- 7 x \cdot (x + 2) = - 7 x^{2} - 14 x$	$(- 7 x^{2} - 2 x + 24) - (- 7 x^{2} - 14 x) = 12 x + 24$
$12 x + 24$	$\frac{12 x}{x}$	$12$	$12 \cdot (x + 2) = 12 x + 24$	$(12 x + 24) - (12 x + 24) = 0$

So the result of the euclidean division is:

x^{2} - 7 x + 12

(x + 2) \cdot (x^{2} - 7 x + 12)

7) Let's factor in

ℝ

(x^{2} - 7 x + 12)

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

Δ = {(-7)}^{2} - 4 \cdot 1 \cdot 12

Δ = 49 - 4 \cdot 12

Δ = 49 - 48

Δ = 1

Δ > 0

, the equation has two solutions:

For the 1st solution, we obtain:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{7 - 1}{2}

x1 = \frac{6}{2}

x1 = 3

Solution second:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{7 + 1}{2}

x2 = \frac{8}{2}

x2 = 4

Factors identified:

(x - 3) \cdot (x - 4)

▷Result: $(x + 1) \cdot (x + 2) \cdot (x - 3) \cdot (x - 4)$