Algebra Examples

${(y - x^{2})}^{3}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(y - x^{2})}^{3}$ , $n = 3$ so the coefficients of the expansion will correspond with line $4$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 3 - 3 - 1$ .

$1 a^{3} b^{0} + 3 a^{2} b + 3 a b^{2} + 1 a^{0} b^{3}$

Step 3

Substitute the actual values of $a$ $y$ and $b$ $- x^{2}$ into the expression.

$1 {(y)}^{3} {(- x^{2})}^{0} + 3 {(y)}^{2} {(- x^{2})}^{1} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply ${(y)}^{3}$ by $1$ .

${(y)}^{3} {(- x^{2})}^{0} + 3 {(y)}^{2} {(- x^{2})}^{1} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.2

Apply the product rule to $- x^{2}$ .

$y^{3} ({(- 1)}^{0} {(x^{2})}^{0}) + 3 {(y)}^{2} {(- x^{2})}^{1} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.3

Rewrite using the commutative property of multiplication.

${(- 1)}^{0} y^{3} {(x^{2})}^{0} + 3 {(y)}^{2} {(- x^{2})}^{1} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.4

Anything raised to $0$ is $1$ .

$1 y^{3} {(x^{2})}^{0} + 3 {(y)}^{2} {(- x^{2})}^{1} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.5

Multiply $y^{3}$ by $1$ .

$y^{3} {(x^{2})}^{0} + 3 {(y)}^{2} {(- x^{2})}^{1} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.6

Multiply the exponents in ${(x^{2})}^{0}$ .

Tap for more steps...

Step 4.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{3} x^{2 \cdot 0} + 3 {(y)}^{2} {(- x^{2})}^{1} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.6.2

Multiply $2$ by $0$ .

$y^{3} x^{0} + 3 {(y)}^{2} {(- x^{2})}^{1} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.7

Anything raised to $0$ is $1$ .

$y^{3} \cdot 1 + 3 {(y)}^{2} {(- x^{2})}^{1} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.8

Multiply $y^{3}$ by $1$ .

$y^{3} + 3 {(y)}^{2} {(- x^{2})}^{1} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.9

Simplify.

$y^{3} + 3 y^{2} (- x^{2}) + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.10

Rewrite using the commutative property of multiplication.

$y^{3} + 3 \cdot - 1 y^{2} x^{2} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.11

Multiply $3$ by $- 1$ .

$y^{3} - 3 y^{2} x^{2} + 3 {(y)}^{1} {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.12

Simplify.

$y^{3} - 3 y^{2} x^{2} + 3 y {(- x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.13

Apply the product rule to $- x^{2}$ .

$y^{3} - 3 y^{2} x^{2} + 3 y ({(- 1)}^{2} {(x^{2})}^{2}) + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.14

Rewrite using the commutative property of multiplication.

$y^{3} - 3 y^{2} x^{2} + 3 \cdot {(- 1)}^{2} y {(x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.15

Raise $- 1$ to the power of $2$ .

$y^{3} - 3 y^{2} x^{2} + 3 \cdot 1 y {(x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.16

Multiply $3$ by $1$ .

$y^{3} - 3 y^{2} x^{2} + 3 y {(x^{2})}^{2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.17

Multiply the exponents in ${(x^{2})}^{2}$ .

Tap for more steps...

Step 4.17.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{3} - 3 y^{2} x^{2} + 3 y x^{2 \cdot 2} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.17.2

Multiply $2$ by $2$ .

$y^{3} - 3 y^{2} x^{2} + 3 y x^{4} + 1 {(y)}^{0} {(- x^{2})}^{3}$

Step 4.18

Multiply ${(y)}^{0}$ by $1$ .

$y^{3} - 3 y^{2} x^{2} + 3 y x^{4} + {(y)}^{0} {(- x^{2})}^{3}$

Step 4.19

Anything raised to $0$ is $1$ .

$y^{3} - 3 y^{2} x^{2} + 3 y x^{4} + 1 {(- x^{2})}^{3}$

Step 4.20

Multiply ${(- x^{2})}^{3}$ by $1$ .

$y^{3} - 3 y^{2} x^{2} + 3 y x^{4} + {(- x^{2})}^{3}$

Step 4.21

Apply the product rule to $- x^{2}$ .

$y^{3} - 3 y^{2} x^{2} + 3 y x^{4} + {(- 1)}^{3} {(x^{2})}^{3}$

Step 4.22

Raise $- 1$ to the power of $3$ .

$y^{3} - 3 y^{2} x^{2} + 3 y x^{4} - {(x^{2})}^{3}$

Step 4.23

Multiply the exponents in ${(x^{2})}^{3}$ .

Tap for more steps...

Step 4.23.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{3} - 3 y^{2} x^{2} + 3 y x^{4} - x^{2 \cdot 3}$

Step 4.23.2

Multiply $2$ by $3$ .

$y^{3} - 3 y^{2} x^{2} + 3 y x^{4} - x^{6}$