factor 16x^3-16x^2-25x+25 — step-by-step solution | Explain MAths

factor 16x^3-16x^2-25x+25

1) Let's factor in

ℝ

:

16 x^{3}

- 16 x^{2}

- 25 x

+ 25

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

x = 1

⇥Let's compute the value of:

16 \cdot 1^{3}

- 16 \cdot 1^{2}

- 25 \cdot 1

+ 25

⇥

16

- 16

- 25

+ 25

⇥ⓘ Start from

16

and move back

16

on the number line, down to zero:

16 - 16

=

0

⇥

0 - 25 + 25

⇥ⓘ Adding 0 changes nothing:

0 - 25

=

-25

■Start from

-25

and move forward

25

on the number line, up to zero:

- 25 + 25

=

0

⇥

0

⇥Root found:

x = 1

⇥1.2) Perform polynomial long division of

(16 x^{3}

- 16 x^{2}

- 25 x

+ 25)

by

(x - 1)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$16 x^{3}$ $- 16 x^{2}$ $- 25 x$ $+ 25$	$\frac{16 x^{3}}{x}$	$16 x^{2}$	$16 x^{2} \cdot (x - 1) = 16 x^{3} - 16 x^{2}$	$(16 x^{3}$ $- 16 x^{2}$ $- 25 x$ $+ 25) - (16 x^{3} - 16 x^{2}) = - 25 x + 25$
$- 25 x + 25$	$- \frac{25 x}{x}$	$-25$	$-25 \cdot (x - 1) = - 25 x + 25$	$(- 25 x + 25) - (- 25 x + 25) = 0$

⇥So the result of the euclidean division is:

16 x^{2} - 25

⇥1.3) Let's factorize in

ℝ

:

(16 x^{2} - 25)

⇥For

16 x^{2} - 25

, we recognize a difference of two squares:

{(4 x)}^{2} - 5^{2}

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = 4 x

and

b = 5

⇥You can factorize the expression:

16 x^{2} - 25

into:

(4 x - 5) \cdot (4 x + 5)

▷The factor

(16 x^{2} - 25)

factors as:

(4 x + 5) \cdot (4 x - 5)

▶Factored expression:

(x - 1) \cdot (4 x + 5) \cdot (4 x - 5)