factor 2x^3+x^2-32x-16 — step-by-step solution

1) Let's factor in

ℝ

2 x^{3}

+ x^{2}

- 32 x

- 16

⇥1.1) Try the obvious solution first:

x = 4

⇥Let's compute the value of:

2 \cdot 4^{3}

+ 4^{2}

- 32 \cdot 4

- 16

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

4^{3}

4 \cdot 4 \cdot 4

64

4^{2}

4 \cdot 4

16

■Multiplications are done before additions and subtractions:

32 \cdot 4

(30 + 2) \cdot 4

30 \cdot 4 + 2 \cdot 4

120 + 8

128

⇥

2 \cdot 64

+ 16

- 128

- 16

⇥ⓘ Multiplications are done before additions and subtractions:

2 \cdot 64

2 \cdot (60 + 4)

2 \cdot 60 + 2 \cdot 4

120 + 8

128

■Start from

16

and move back

128

on the number line, going below zero:

16 - 128

-112

⇥

128 - 112 - 16

⇥ⓘ Start from

128

and move back

112

on the number line:

128 - 112

16

⇥

16 - 16

⇥ⓘ Start from

16

and move back

16

on the number line, down to zero:

16 - 16

0

⇥

0

⇥Root found:

x = 4

⇥1.2) Perform polynomial long division of

(2 x^{3}

+ x^{2}

- 32 x

- 16)

(x - 4)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$2 x^{3}$ $+ x^{2}$ $- 32 x$ $- 16$	$\frac{2 x^{3}}{x}$	$2 x^{2}$	$2 x^{2} \cdot (x - 4) = 2 x^{3} - 8 x^{2}$	$(2 x^{3}$ $+ x^{2}$ $- 32 x$ $- 16) - (2 x^{3} - 8 x^{2}) = 9 x^{2} - 32 x - 16$
$9 x^{2} - 32 x - 16$	$\frac{9 x^{2}}{x}$	$9 x$	$9 x \cdot (x - 4) = 9 x^{2} - 36 x$	$(9 x^{2} - 32 x - 16) - (9 x^{2} - 36 x) = 4 x - 16$
$4 x - 16$	$\frac{4 x}{x}$	$4$	$4 \cdot (x - 4) = 4 x - 16$	$(4 x - 16) - (4 x - 16) = 0$

⇥So the result of the euclidean division is:

2 x^{2} + 9 x + 4

⇥1.3) Let's factor in

ℝ

(2 x^{2} + 9 x + 4)

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = 9

and

c = 4

⇥

Δ = 9^{2} - 4 \cdot 2 \cdot 4

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

9^{2}

9 \cdot 9

81

■Multiplications are done before additions and subtractions:

2 \cdot 4

8

⇥

Δ = 81 - 8 \cdot 4

⇥ⓘ Multiplications are done before additions and subtractions:

8 \cdot 4

32

⇥

Δ = 81 - 32

⇥ⓘ Start from

81

and move back

32

on the number line:

81 - 32

49

⇥

Δ = 49

⇥Since

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 9 - \sqrt{49}}{2 \cdot 2}

⇥

x_{1} = \frac{- 9 - \sqrt{49}}{2 \cdot 2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{49}

7

;

7^{2}

49

⇥ⅰ

2 \cdot 2

4

⇥

x_{1} = \frac{- 9 - 7}{4}

⇥ⓘ The numerator and the denominator are computed before the division:

- 9 - 7

-16

⇥

x_{1} = - \frac{16}{4}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

4

, their greatest common divisor:

- \frac{16}{4}

-4

⇥

x_{1} = -4

⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 9 + \sqrt{49}}{2 \cdot 2}

⇥

x_{2} = \frac{- 9 + \sqrt{49}}{2 \cdot 2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{49}

7

;

7^{2}

49

⇥ⅰ

2 \cdot 2

4

⇥

x_{2} = \frac{- 9 + 7}{4}

⇥ⓘ The numerator and the denominator are computed before the division:

- 9 + 7

-2

⇥

x_{2} = - \frac{2}{4}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{2}{4}

- \frac{1 \cdot 2}{2 \cdot 2}

- \frac{1}{2}

⇥

x_{2} = - \frac{1}{2}

⇥A trinomial with roots

x_{1}

and

x_{2}

a \cdot (x - x_{1}) \cdot (x - x_{2})

; bringing the coefficient into the brackets:

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

(x + 4) \cdot (2 x + 1)

▷The factor

(2 x^{2} + 9 x + 4)

factors as:

(x + 4) \cdot (2 x + 1)

▶Factored expression:

(x + 4) \cdot (x - 4) \cdot (2 x + 1)