factor 3x^3+23x^2+40x-16 — step-by-step solution

⚠Ⓘ 3x is read as the product 3*x: for a power, write 3^x‖23x is read as the product 23*x: for a power, write 23^x‖40x is read as the product 40*x: for a power, write 40^x

1) Let's factor in

ℝ

3 x^{3}

+ 23 x^{2}

+ 40 x

- 16

Let's start with the obvious candidate:

x = \frac{1}{3}

2) Let's evaluate the value of:

3 \cdot {(\frac{1}{3})}^{3}

+ 23 \cdot {(\frac{1}{3})}^{2}

+ 40 \cdot \frac{1}{3}

- 16

ℹ

40 \cdot \frac{1}{3}

\frac{40}{3}

‖

\frac{40}{3} - 16

\frac{40 - 16 \cdot 3}{3}

3 \cdot {(\frac{1}{3})}^{3} + 23 \cdot {(\frac{1}{3})}^{2} + \frac{40 - 16 \cdot 3}{3}

3 \cdot {(\frac{1}{3})}^{3} + 23 \cdot {(\frac{1}{3})}^{2} + \frac{40 - 48}{3}

3 \cdot {(\frac{1}{3})}^{3} + 23 \cdot {(\frac{1}{3})}^{2} - \frac{8}{3}

ℹ

{(\frac{1}{3})}^{3}

\frac{1}{3^{3}}

‖

{(\frac{1}{3})}^{2}

\frac{1}{3^{2}}

3 \cdot \frac{1}{3^{3}} + 23 \cdot \frac{1}{3^{2}} - \frac{8}{3}

ℹ

3^{3}

27

‖

3^{2}

9

3 \cdot \frac{1}{27} + 23 \cdot \frac{1}{9} - \frac{8}{3}

ℹ

3 \cdot \frac{1}{27}

\frac{3}{27}

‖

23 \cdot \frac{1}{9}

\frac{23}{9}

‖

\frac{3}{27} + \frac{23}{9}

\frac{3 + 23 \cdot 3}{27}

\frac{3 + 23 \cdot 3}{27} - \frac{8}{3}

\frac{3 + 69}{27} - \frac{8}{3}

\frac{72}{27} - \frac{8}{3}

\frac{72 - 8 \cdot 9}{27}

\frac{72 - 72}{27}

\frac{0}{27}

0

Factor found:

x = \frac{1}{3}

3) Perform polynomial long division of

3 x^{3}

+ 23 x^{2}

+ 40 x

- 16

x - \frac{1}{3}

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$3 x^{3}$ $+ 23 x^{2}$ $+ 40 x$ $- 16$	$\frac{3 x^{3}}{x}$	$3 x^{2}$	$3 x^{2} \cdot (x - \frac{1}{3}) = 3 x^{3} - x^{2}$	$(3 x^{3}$ $+ 23 x^{2}$ $+ 40 x$ $- 16) - (3 x^{3} - x^{2}) = 24 x^{2} + 40 x - 16$
$24 x^{2} + 40 x - 16$	$\frac{24 x^{2}}{x}$	$24 x$	$24 x \cdot (x - \frac{1}{3}) = 24 x^{2} - 8 x$	$(24 x^{2} + 40 x - 16) - (24 x^{2} - 8 x) = 48 x - 16$
$48 x - 16$	$\frac{48 x}{x}$	$48$	$48 \cdot (x - \frac{1}{3}) = 48 x - 16$	$(48 x - 16) - (48 x - 16) = 0$

Therefore, the result of the euclidean division is:

3 x^{2} + 24 x + 48

(x - \frac{1}{3}) \cdot (3 x^{2} + 24 x + 48)

4) Let's factorize in

ℝ

(3 x^{2} + 24 x + 48)

You can factorize this expression into:

3 \cdot (x^{2} + 8 x + 16)

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

Δ = 8^{2} - 4 \cdot 1 \cdot 16

Δ = 64 - 4 \cdot 16

Δ = 64 - 64

Δ = 0

Since

Δ = 0

, the equation has one solution:

For the unique solution, we obtain:

- \frac{b}{2 a}

x = - \frac{8}{2}

x = -4

Factor found:

{(x + 4)}^{2}

▶Factored expression:

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