Algebra Examples

${(r - t)}^{4}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(r)}^{4 - k} \cdot {(- t)}^{k}$

Step 2

Expand the summation.

$\frac{4!}{(4 - 0)! 0!} {(r)}^{4 - 0} \cdot {(- t)}^{0} + \frac{4!}{(4 - 1)! 1!} {(r)}^{4 - 1} \cdot {(- t)}^{1} + \frac{4!}{(4 - 2)! 2!} {(r)}^{4 - 2} \cdot {(- t)}^{2} + \frac{4!}{(4 - 3)! 3!} {(r)}^{4 - 3} \cdot {(- t)}^{3} + \frac{4!}{(4 - 4)! 4!} {(r)}^{4 - 4} \cdot {(- t)}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(r)}^{4} \cdot {(- t)}^{0} + 4 \cdot {(r)}^{3} \cdot {(- t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(r)}^{4}$ by $1$ .

${(r)}^{4} \cdot {(- t)}^{0} + 4 \cdot {(r)}^{3} \cdot {(- t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.2

Apply the product rule to $- t$ .

$r^{4} \cdot ({(- 1)}^{0} t^{0}) + 4 \cdot {(r)}^{3} \cdot {(- t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.3

Rewrite using the commutative property of multiplication.

${(- 1)}^{0} r^{4} t^{0} + 4 \cdot {(r)}^{3} \cdot {(- t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.4

Anything raised to $0$ is $1$ .

$1 r^{4} t^{0} + 4 \cdot {(r)}^{3} \cdot {(- t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.5

Multiply $r^{4}$ by $1$ .

$r^{4} t^{0} + 4 \cdot {(r)}^{3} \cdot {(- t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.6

Anything raised to $0$ is $1$ .

$r^{4} \cdot 1 + 4 \cdot {(r)}^{3} \cdot {(- t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.7

Multiply $r^{4}$ by $1$ .

$r^{4} + 4 \cdot {(r)}^{3} \cdot {(- t)}^{1} + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.8

Simplify.

$r^{4} + 4 r^{3} \cdot (- t) + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.9

Rewrite using the commutative property of multiplication.

$r^{4} + 4 \cdot - 1 r^{3} t + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.10

Multiply $4$ by $- 1$ .

$r^{4} - 4 r^{3} t + 6 \cdot {(r)}^{2} \cdot {(- t)}^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.11

Apply the product rule to $- t$ .

$r^{4} - 4 r^{3} t + 6 r^{2} \cdot ({(- 1)}^{2} t^{2}) + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.12

Rewrite using the commutative property of multiplication.

$r^{4} - 4 r^{3} t + 6 \cdot {(- 1)}^{2} r^{2} t^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.13

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

$r^{4} - 4 r^{3} t + 6 \cdot 1 r^{2} t^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.14

Multiply $6$ by $1$ .

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} + 4 \cdot {(r)}^{1} \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.15

Simplify.

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} + 4 \cdot r \cdot {(- t)}^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.16

Apply the product rule to $- t$ .

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} + 4 r \cdot ({(- 1)}^{3} t^{3}) + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.17

Rewrite using the commutative property of multiplication.

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} + 4 \cdot {(- 1)}^{3} r t^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.18

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

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} + 4 \cdot - 1 r t^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.19

Multiply $4$ by $- 1$ .

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} - 4 r t^{3} + 1 \cdot {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.20

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

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} - 4 r t^{3} + {(r)}^{0} \cdot {(- t)}^{4}$

Step 4.21

Anything raised to $0$ is $1$ .

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} - 4 r t^{3} + 1 \cdot {(- t)}^{4}$

Step 4.22

Multiply ${(- t)}^{4}$ by $1$ .

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} - 4 r t^{3} + {(- t)}^{4}$

Step 4.23

Apply the product rule to $- t$ .

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} - 4 r t^{3} + {(- 1)}^{4} t^{4}$

Step 4.24

Raise $- 1$ to the power of $4$ .

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} - 4 r t^{3} + 1 t^{4}$

Step 4.25

Multiply $t^{4}$ by $1$ .

$r^{4} - 4 r^{3} t + 6 r^{2} t^{2} - 4 r t^{3} + t^{4}$