factor r^3-9r^2+15r+25 — step-by-step solution

1) Let's factor in

ℝ

r^{3}

- 9 r^{2}

+ 15 r

+ 25

⇥1.1) Try the obvious solution first:

r = -1

⇥Let's compute the value of:

{(-1)}^{3}

- 9 \cdot {(-1)}^{2}

+ 15 \cdot (-1)

+ 25

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

{(-1)}^{3}

(-1) \cdot (-1) \cdot (-1)

-1

{(-1)}^{2}

(-1) \cdot (-1)

1

■Multiplying by 1 changes nothing:

9 \cdot 1

9

■Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

15 \cdot (-1)

-15

⇥

- 1

- 9

- 15

+ 25

⇥ⓘ Start from

-1

and move back

9

more on the number line, away from zero:

- 1 - 9

-10

⇥

- 10 - 15 + 25

⇥ⓘ Start from

-10

and move back

15

more on the number line, away from zero:

- 10 - 15

-25

⇥

- 25 + 25

⇥ⓘ Start from

-25

and move forward

25

on the number line, up to zero:

- 25 + 25

0

⇥

0

⇥Root found:

r = -1

⇥1.2) Compute the quotient and remainder of

(r^{3}

- 9 r^{2}

+ 15 r

+ 25)

(r + 1)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$r^{3}$ $- 9 r^{2}$ $+ 15 r$ $+ 25$	$\frac{r^{3}}{r}$	$r^{2}$	$r^{2} \cdot (r + 1) = r^{3} + r^{2}$	$(r^{3}$ $- 9 r^{2}$ $+ 15 r$ $+ 25) - (r^{3} + r^{2}) = - 10 r^{2} + 15 r + 25$
$- 10 r^{2} + 15 r + 25$	$- \frac{10 r^{2}}{r}$	$- 10 r$	$- 10 r \cdot (r + 1) = - 10 r^{2} - 10 r$	$(- 10 r^{2} + 15 r + 25) - (- 10 r^{2} - 10 r) = 25 r + 25$
$25 r + 25$	$\frac{25 r}{r}$	$25$	$25 \cdot (r + 1) = 25 r + 25$	$(25 r + 25) - (25 r + 25) = 0$

⇥Therefore, the result of the euclidean division is:

r^{2} - 10 r + 25

⇥1.3) Let's factor in

ℝ

(r^{2} - 10 r + 25)

⇥You can rewrite the expression as:

{(r - 5)}^{2}

▷The factor

(r^{2} - 10 r + 25)

factors as:

{(r - 5)}^{2}

▶Factored expression:

(r + 1) \cdot {(r - 5)}^{2}