factor r^3-25r^2-9r+225 — step-by-step solution

1) Let's factorize in

ℝ

r^{3}

- 25 r^{2}

- 9 r

+ 225

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

r = 3

⇥Let's compute the value of:

3^{3}

- 25 \cdot 3^{2}

- 9 \cdot 3

+ 225

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

3^{3}

3 \cdot 3 \cdot 3

27

3^{2}

3 \cdot 3

9

■Multiplications are done before additions and subtractions:

9 \cdot 3

27

⇥

27

- 25 \cdot 9

- 27

+ 225

⇥ⓘ Multiplications are done before additions and subtractions:

25 \cdot 9

(20 + 5) \cdot 9

20 \cdot 9 + 5 \cdot 9

180 + 45

225

⇥

27

- 225

- 27

+ 225

⇥ⓘ Start from

27

and move back

225

on the number line, going below zero:

27 - 225

-198

⇥

- 198 - 27 + 225

⇥ⓘ Start from

-198

and move back

27

more on the number line, away from zero:

- 198 - 27

-225

⇥

- 225 + 225

⇥ⓘ Start from

-225

and move forward

225

on the number line, up to zero:

- 225 + 225

0

⇥

0

⇥Root found:

r = 3

⇥1.2) Perform polynomial long division of

(r^{3}

- 25 r^{2}

- 9 r

+ 225)

(r - 3)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$r^{3}$ $- 25 r^{2}$ $- 9 r$ $+ 225$	$\frac{r^{3}}{r}$	$r^{2}$	$r^{2} \cdot (r - 3) = r^{3} - 3 r^{2}$	$(r^{3}$ $- 25 r^{2}$ $- 9 r$ $+ 225) - (r^{3} - 3 r^{2}) = - 22 r^{2} - 9 r + 225$
$- 22 r^{2} - 9 r + 225$	$- \frac{22 r^{2}}{r}$	$- 22 r$	$- 22 r \cdot (r - 3) = - 22 r^{2} + 66 r$	$(- 22 r^{2} - 9 r + 225) - (- 22 r^{2} + 66 r) = - 75 r + 225$
$- 75 r + 225$	$- \frac{75 r}{r}$	$-75$	$-75 \cdot (r - 3) = - 75 r + 225$	$(- 75 r + 225) - (- 75 r + 225) = 0$

⇥So the result of the euclidean division is:

r^{2} - 22 r - 75

⇥1.3) Let's factor in

ℝ

(r^{2} - 22 r - 75)

⇥Given the quadratic form

a r^{2} + b r + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -22

and

c = -75

⇥

Δ = {(-22)}^{2} - 4 \cdot 1 \cdot (-75)

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

{(-22)}^{2}

(-22) \cdot (-22)

484

⇥

Δ = 484 - 4 \cdot (-75)

⇥ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-75)

4 \cdot 75

4 \cdot (70 + 5)

4 \cdot 70 + 4 \cdot 5

280 + 20

300

⇥

Δ = 484 + 300

⇥ⓘ Start from

484

and move forward

300

on the number line:

484 + 300

784

⇥

Δ = 784

⇥As

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

r_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-22) - \sqrt{784}}{2 \cdot 1}

⇥

r_{1} = \frac{22 - \sqrt{784}}{2}

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

\sqrt{784}

28

;

28^{2}

784

⇥

r_{1} = \frac{22 - 28}{2}

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

22 - 28

-6

⇥

r_{1} = - \frac{6}{2}

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

2

, their greatest common divisor:

- \frac{6}{2}

-3

⇥

r_{1} = -3

⇥The quadratic formula gives

r_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-22) + \sqrt{784}}{2 \cdot 1}

⇥

r_{2} = \frac{22 + \sqrt{784}}{2}

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

\sqrt{784}

28

;

28^{2}

784

⇥

r_{2} = \frac{22 + 28}{2}

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

22 + 28

50

⇥

r_{2} = \frac{50}{2}

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

2

, their greatest common divisor:

\frac{50}{2}

25

⇥

r_{2} = 25

▷The factor

(r^{2} - 22 r - 75)

factors as:

(r + 3) \cdot (r - 25)

▶Factored expression:

(r + 3) \cdot (r - 3) \cdot (r - 25)