factor x^3-8x^2+17x-10 — step-by-step solution

1) Let's factor in

ℝ

x^{3}

- 8 x^{2}

+ 17 x

- 10

Try the obvious solution first:

x = 1

2) Let's evaluate the value of:

1^{3}

- 8 \cdot 1^{2}

+ 17 \cdot 1

- 10

- 9 - 8 + 17

- 17 + 17

0

Factor found:

x = 1

3) Perform polynomial long division:

x^{3}

- 8 x^{2}

+ 17 x

- 10

x - 1

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- 8 x^{2}$ $+ 17 x$ $- 10$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 1) = x^{3} - x^{2}$	$(x^{3}$ $- 8 x^{2}$ $+ 17 x$ $- 10) - (x^{3} - x^{2}) = - 7 x^{2} + 17 x - 10$
$- 7 x^{2} + 17 x - 10$	$- \frac{7 x^{2}}{x}$	$- 7 x$	$- 7 x \cdot (x - 1) = - 7 x^{2} + 7 x$	$(- 7 x^{2} + 17 x - 10) - (- 7 x^{2} + 7 x) = 10 x - 10$
$10 x - 10$	$\frac{10 x}{x}$	$10$	$10 \cdot (x - 1) = 10 x - 10$	$(10 x - 10) - (10 x - 10) = 0$

So the result of the euclidean division is:

x^{2} - 7 x + 10

(x - 1) \cdot (x^{2} - 7 x + 10)

4) Let's factorize in

ℝ

(x^{2} - 7 x + 10)

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 10

Δ = 49 - 4 \cdot 10

Δ = 49 - 40

Δ = 9

Δ > 0

, the equation has two solutions:

Solution 1st:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{7 - \sqrt{9}}{2}

x1 = \frac{7 - 3}{2}

x1 = \frac{4}{2}

x1 = 2

Solution second:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{7 + \sqrt{9}}{2}

x2 = \frac{7 + 3}{2}

x2 = \frac{10}{2}

x2 = 5

Factors found:

(x - 2) \cdot (x - 5)

▶Factored form:

(x - 1) \cdot (x - 2) \cdot (x - 5)