factor 8x^3-62x^2+43x-7 — step-by-step solution

1) Let's factorize in

ℝ

8 x^{3}

- 62 x^{2}

+ 43 x

- 7

⇥1.1) Let's start with the obvious candidate:

x = 7

⇥Let's evaluate the value of:

8 \cdot 7^{3}

- 62 \cdot 7^{2}

+ 43 \cdot 7

- 7

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

7^{3}

7 \cdot 7 \cdot 7

343

7^{2}

7 \cdot 7

49

■Multiplications are done before additions and subtractions:

43 \cdot 7

(40 + 3) \cdot 7

40 \cdot 7 + 3 \cdot 7

280 + 21

301

⇥

8 \cdot 343

- 62 \cdot 49

+ 301

- 7

⇥ⓘ Multiplications are done before additions and subtractions:

8 \cdot 343

8 \cdot (300 + 40 + 3)

8 \cdot 300 + 8 \cdot 40 + 8 \cdot 3

2400 + 320 + 24

2744

62 \cdot 49

62 \cdot (40 + 9)

62 \cdot 40 + 62 \cdot 9

2480 + 558

3038

■Start from

301

and move back

7

on the number line:

301 - 7

294

⇥

2744 - 3038 + 294

⇥ⓘ Start from

2744

and move back

3038

on the number line, going below zero:

2744 - 3038

-294

⇥

- 294 + 294

⇥ⓘ Start from

-294

and move forward

294

on the number line, up to zero:

- 294 + 294

0

⇥

0

⇥Root found:

x = 7

⇥1.2) Let's start the euclidean division of

(8 x^{3}

- 62 x^{2}

+ 43 x

- 7)

(x - 7)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$8 x^{3}$ $- 62 x^{2}$ $+ 43 x$ $- 7$	$\frac{8 x^{3}}{x}$	$8 x^{2}$	$8 x^{2} \cdot (x - 7) = 8 x^{3} - 56 x^{2}$	$(8 x^{3}$ $- 62 x^{2}$ $+ 43 x$ $- 7) - (8 x^{3} - 56 x^{2}) = - 6 x^{2} + 43 x - 7$
$- 6 x^{2} + 43 x - 7$	$- \frac{6 x^{2}}{x}$	$- 6 x$	$- 6 x \cdot (x - 7) = - 6 x^{2} + 42 x$	$(- 6 x^{2} + 43 x - 7) - (- 6 x^{2} + 42 x) = x - 7$
$x - 7$	$\frac{x}{x}$	$1$	$x - 7 = x - 7$	$(x - 7) - (x - 7) = 0$

⇥Therefore, the result of the euclidean division is:

8 x^{2} - 6 x + 1

⇥1.3) Let's factorize in

ℝ

(8 x^{2} - 6 x + 1)

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 8

b = -6

and

c = 1

⇥

Δ = {(-6)}^{2} - 4 \cdot 8 \cdot 1

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

{(-6)}^{2}

(-6) \cdot (-6)

36

⇥

Δ = 36 - 4 \cdot 8

⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 8

32

⇥

Δ = 36 - 32

⇥ⓘ Start from

36

and move back

32

on the number line:

36 - 32

4

⇥

Δ = 4

⇥As

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-6) - \sqrt{4}}{2 \cdot 8}

⇥

x_{1} = \frac{6 - \sqrt{4}}{2 \cdot 8}

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

\sqrt{4}

2

;

2^{2}

4

⇥ⅰ

2 \cdot 8

16

⇥

x_{1} = \frac{6 - 2}{16}

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

6 - 2

4

⇥

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

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

4

, their greatest common divisor:

\frac{4}{16}

\frac{1 \cdot 4}{4 \cdot 4}

\frac{1}{4}

⇥

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

⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-6) + \sqrt{4}}{2 \cdot 8}

⇥

x_{2} = \frac{6 + \sqrt{4}}{2 \cdot 8}

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

\sqrt{4}

2

;

2^{2}

4

⇥ⅰ

2 \cdot 8

16

⇥

x_{2} = \frac{6 + 2}{16}

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

6 + 2

8

⇥

x_{2} = \frac{8}{16}

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

8

, their greatest common divisor:

\frac{8}{16}

\frac{1 \cdot 8}{2 \cdot 8}

\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:

8 \cdot (x - \frac{1}{2}) \cdot (x - \frac{1}{4})

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

▷The factor

(8 x^{2} - 6 x + 1)

factors as:

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

▶Factored expression:

(x - 7) \cdot (2 x - 1) \cdot (4 x - 1)