factor 2x^3+7x^2-10x-24 — step-by-step solution

⚠Ⓘ 2x is read as the product 2*x: for a power, write 2^x‖7x is read as the product 7*x: for a power, write 7^x‖10x is read as the product 10*x: for a power, write 10^x

1) Let's factor in

ℝ

2 x^{3}

+ 7 x^{2}

- 10 x

- 24

Let's start with the obvious candidate:

x = - \frac{3}{2}

2) Let's evaluate the value of:

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

+ 7 \cdot {(- \frac{3}{2})}^{2}

- 10 \cdot (- \frac{3}{2})

- 24

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

+ 7 \cdot {(- \frac{3}{2})}^{2}

+ 10 \cdot \frac{3}{2}

- 24

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

+ 7 \cdot {(- \frac{3}{2})}^{2}

+ \frac{30}{2}

- 24

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

+ 7 \cdot {(- \frac{3}{2})}^{2}

+ 15

- 24

2 \cdot {(- \frac{3}{2})}^{3} + 7 \cdot {(- \frac{3}{2})}^{2} - 9

ℹ

{(- \frac{3}{2})}^{3}

\frac{{(-3)}^{3}}{2^{3}}

‖

{(- \frac{3}{2})}^{2}

\frac{{(-3)}^{2}}{2^{2}}

2 \cdot \frac{{(-3)}^{3}}{2^{3}} + 7 \cdot \frac{{(-3)}^{2}}{2^{2}} - 9

ℹ

{(-3)}^{3}

-27

‖

2^{3}

8

‖

{(-3)}^{2}

9

‖

2^{2}

4

2 \cdot (- \frac{27}{8}) + 7 \cdot \frac{9}{4} - 9

- 2 \cdot \frac{27}{8} + \frac{63}{4} - 9

ℹ

- 2 \cdot \frac{27}{8}

- \frac{54}{8}

‖

\frac{63}{4} - 9

\frac{63 - 9 \cdot 4}{4}

- \frac{54}{8} + \frac{63 - 9 \cdot 4}{4}

ℹ

- \frac{54}{8}

- \frac{27}{4}

‖

-9 \cdot 4

-36

- \frac{27}{4} + \frac{63 - 36}{4}

- \frac{27}{4} + \frac{27}{4}

\frac{- 27 + 27}{4}

\frac{0}{4}

0

Factor identified:

x = - \frac{3}{2}

3) Let's start the euclidean division of

2 x^{3}

+ 7 x^{2}

- 10 x

- 24

x - (\frac{-3}{2})

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$2 x^{3}$ $+ 7 x^{2}$ $- 10 x$ $- 24$	$\frac{2 x^{3}}{x}$	$2 x^{2}$	$2 x^{2} \cdot (x - (\frac{-3}{2})) = 2 x^{3} + 3 x^{2}$	$(2 x^{3}$ $+ 7 x^{2}$ $- 10 x$ $- 24) - (2 x^{3} + 3 x^{2}) = 4 x^{2} - 10 x - 24$
$4 x^{2} - 10 x - 24$	$\frac{4 x^{2}}{x}$	$4 x$	$4 x \cdot (x - (\frac{-3}{2})) = 4 x^{2} + 6 x$	$(4 x^{2} - 10 x - 24) - (4 x^{2} + 6 x) = - 16 x - 24$
$- 16 x - 24$	$- \frac{16 x}{x}$	$-16$	$-16 \cdot (x - (\frac{-3}{2})) = - 16 x - 24$	$(- 16 x - 24) - (- 16 x - 24) = 0$

Therefore, the result of the euclidean division is:

2 x^{2} + 4 x - 16

(x - (\frac{-3}{2})) \cdot (2 x^{2} + 4 x - 16)

(x + \frac{3}{2}) \cdot (2 x^{2} + 4 x - 16)

4) Let's factorize in

ℝ

(2 x^{2} + 4 x - 16)

This expression can be factorized into:

2 \cdot (x^{2} + 2 x - 8)

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

Δ = 2^{2} - 4 \cdot 1 \cdot (-8)

Δ = 4 - 4 \cdot (-8)

Δ = 4 + 32

Δ = 36

Since

Δ > 0

, the equation has two solutions:

The first solution is:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{- 2 - \sqrt{36}}{2}

x1 = \frac{- 2 - 6}{2}

x1 = - \frac{8}{2}

x1 = -4

The second solution is:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{- 2 + \sqrt{36}}{2}

x2 = \frac{- 2 + 6}{2}

x2 = \frac{4}{2}

x2 = 2

Factors found:

(x + 4) \cdot (x - 2)

▶Factored expression:

2 \cdot (x + 4) \cdot (x - 2) \cdot (x + \frac{3}{2})