Simplify (4a^3-4(a+h)^3)/h

$\frac{4 a^{3} - 4 {(a + h)}^{3}}{h}$

Step 1

Simplify the numerator.

Factor

$4$ out of

$4 a^{3} - 4 {(a + h)}^{3}$ .

Factor

$4$ out of

$4 a^{3}$ .

$\frac{4 (a^{3}) - 4 {(a + h)}^{3}}{h}$

Factor

$4$ out of

$- 4 {(a + h)}^{3}$ .

$\frac{4 (a^{3}) + 4 (- {(a + h)}^{3})}{h}$

Factor

$4$ out of

$4 (a^{3}) + 4 (- {(a + h)}^{3})$ .

$\frac{4 (a^{3} - {(a + h)}^{3})}{h}$

Since both terms are perfect cubes, factor using the difference of cubes formula,

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

$a = a$ and

$b = a + h$ .

$\frac{4 ((a - (a + h)) (a^{2} + a (a + h) + {(a + h)}^{2}))}{h}$

Simplify.

Apply the distributive property.

$\frac{4 ((a - a - h) (a^{2} + a (a + h) + {(a + h)}^{2}))}{h}$

Subtract

$a$ from

$a$ .

$\frac{4 ((0 - h) (a^{2} + a (a + h) + {(a + h)}^{2}))}{h}$

Subtract

$h$ from

$0$ .

$\frac{4 (- h (a^{2} + a (a + h) + {(a + h)}^{2}))}{h}$

Apply the distributive property.

$\frac{4 (- h (a^{2} + a \cdot a + a h + {(a + h)}^{2}))}{h}$

Multiply

$a$ by

$a$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + {(a + h)}^{2}))}{h}$

Rewrite

${(a + h)}^{2}$ as

$(a + h) (a + h)$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + (a + h) (a + h)))}{h}$

Expand

$(a + h) (a + h)$ using the FOIL Method.

Apply the distributive property.

$\frac{4 (- h (a^{2} + a^{2} + a h + a (a + h) + h (a + h)))}{h}$

Apply the distributive property.

$\frac{4 (- h (a^{2} + a^{2} + a h + a \cdot a + a h + h (a + h)))}{h}$

Apply the distributive property.

$\frac{4 (- h (a^{2} + a^{2} + a h + a \cdot a + a h + h a + h \cdot h))}{h}$

Simplify and combine like terms.

Simplify each term.

Multiply

$a$ by

$a$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + a^{2} + a h + h a + h \cdot h))}{h}$

Multiply

$h$ by

$h$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + a^{2} + a h + h a + h^{2}))}{h}$

Add

$a h$ and

$h a$ .

Reorder

$h$ and

$a$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + a^{2} + a h + a h + h^{2}))}{h}$

Add

$a h$ and

$a h$ .

$\frac{4 (- h (a^{2} + a^{2} + a h + a^{2} + 2 a h + h^{2}))}{h}$

Add

$a^{2}$ and

$a^{2}$ .

$\frac{4 (- h (2 a^{2} + a h + a^{2} + 2 a h + h^{2}))}{h}$

Add

$2 a^{2}$ and

$a^{2}$ .

$\frac{4 (- h (3 a^{2} + a h + 2 a h + h^{2}))}{h}$

Add

$a h$ and

$2 a h$ .

$\frac{4 (- h (3 a^{2} + 3 a h + h^{2}))}{h}$

$\frac{4 \cdot - 1 h (3 a^{2} + 3 a h + h^{2})}{h}$

Combine exponents.

Factor out negative.

$\frac{- (4 h (3 a^{2} + 3 a h + h^{2}))}{h}$

Multiply

$4$ by

$- 1$ .

$\frac{- 4 (h (3 a^{2} + 3 a h + h^{2}))}{h}$

$\frac{- 4 h (3 a^{2} + 3 a h + h^{2})}{h}$

Step 2

Simplify terms.

Cancel the common factor of

$h$ .

Cancel the common factor.

$\frac{- 4 h (3 a^{2} + 3 a h + h^{2})}{h}$

Divide

$- 4 (3 a^{2} + 3 a h + h^{2})$ by

$1$ .

$- 4 (3 a^{2} + 3 a h + h^{2})$

Apply the distributive property.

$- 4 (3 a^{2}) - 4 (3 a h) - 4 h^{2}$

Step 3

Simplify.

Multiply

$3$ by

$- 4$ .

$- 12 a^{2} - 4 (3 a h) - 4 h^{2}$

Multiply

$3$ by

$- 4$ .

$- 12 a^{2} - 12 (a h) - 4 h^{2}$

$- 12 a^{2} - 12 a h - 4 h^{2}$

Simplify (4a^3-4(a+h)^3)/h

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