Pre-Algebra Examples

20 {(r + t)}^{3}

$20 {(r + t)}^{3}$ ,

16 {(r + t)}^{6}

$16 {(r + t)}^{6}$

Step 1

Simplify each term.

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Use the Binomial Theorem.

20 (r^{3} + 3 r^{2} t + 3 r t^{2} + t^{3}), 16 {(r + t)}^{6}

$20 (r^{3} + 3 r^{2} t + 3 r t^{2} + t^{3}), 16 {(r + t)}^{6}$

Apply the distributive property.

20 r^{3} + 20 (3 r^{2} t) + 20 (3 r t^{2}) + 20 t^{3}, 16 {(r + t)}^{6}

$20 r^{3} + 20 (3 r^{2} t) + 20 (3 r t^{2}) + 20 t^{3}, 16 {(r + t)}^{6}$

Simplify.

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Multiply

3

$3$ by

20

$20$ .

20 r^{3} + 60 (r^{2} t) + 20 (3 r t^{2}) + 20 t^{3}, 16 {(r + t)}^{6}

$20 r^{3} + 60 (r^{2} t) + 20 (3 r t^{2}) + 20 t^{3}, 16 {(r + t)}^{6}$

Multiply

3

$3$ by

20

$20$ .

20 r^{3} + 60 (r^{2} t) + 60 (r t^{2}) + 20 t^{3}, 16 {(r + t)}^{6}

$20 r^{3} + 60 (r^{2} t) + 60 (r t^{2}) + 20 t^{3}, 16 {(r + t)}^{6}$

20 r^{3} + 60 (r^{2} t) + 60 (r t^{2}) + 20 t^{3}, 16 {(r + t)}^{6}

$20 r^{3} + 60 (r^{2} t) + 60 (r t^{2}) + 20 t^{3}, 16 {(r + t)}^{6}$

Remove parentheses.

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 {(r + t)}^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 {(r + t)}^{6}$

Use the Binomial Theorem.

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 (r^{6} + 6 r^{5} t + 15 r^{4} t^{2} + 20 r^{3} t^{3} + 15 r^{2} t^{4} + 6 r t^{5} + t^{6})

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 (r^{6} + 6 r^{5} t + 15 r^{4} t^{2} + 20 r^{3} t^{3} + 15 r^{2} t^{4} + 6 r t^{5} + t^{6})$

Apply the distributive property.

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 16 (6 r^{5} t) + 16 (15 r^{4} t^{2}) + 16 (20 r^{3} t^{3}) + 16 (15 r^{2} t^{4}) + 16 (6 r t^{5}) + 16 t^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 16 (6 r^{5} t) + 16 (15 r^{4} t^{2}) + 16 (20 r^{3} t^{3}) + 16 (15 r^{2} t^{4}) + 16 (6 r t^{5}) + 16 t^{6}$

Simplify.

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Multiply

6

$6$ by

16

$16$ .

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 16 (15 r^{4} t^{2}) + 16 (20 r^{3} t^{3}) + 16 (15 r^{2} t^{4}) + 16 (6 r t^{5}) + 16 t^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 16 (15 r^{4} t^{2}) + 16 (20 r^{3} t^{3}) + 16 (15 r^{2} t^{4}) + 16 (6 r t^{5}) + 16 t^{6}$

Multiply

15

$15$ by

16

$16$ .

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 240 (r^{4} t^{2}) + 16 (20 r^{3} t^{3}) + 16 (15 r^{2} t^{4}) + 16 (6 r t^{5}) + 16 t^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 240 (r^{4} t^{2}) + 16 (20 r^{3} t^{3}) + 16 (15 r^{2} t^{4}) + 16 (6 r t^{5}) + 16 t^{6}$

Multiply

20

$20$ by

16

$16$ .

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 240 (r^{4} t^{2}) + 320 (r^{3} t^{3}) + 16 (15 r^{2} t^{4}) + 16 (6 r t^{5}) + 16 t^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 240 (r^{4} t^{2}) + 320 (r^{3} t^{3}) + 16 (15 r^{2} t^{4}) + 16 (6 r t^{5}) + 16 t^{6}$

Multiply

15

$15$ by

16

$16$ .

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 240 (r^{4} t^{2}) + 320 (r^{3} t^{3}) + 240 (r^{2} t^{4}) + 16 (6 r t^{5}) + 16 t^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 240 (r^{4} t^{2}) + 320 (r^{3} t^{3}) + 240 (r^{2} t^{4}) + 16 (6 r t^{5}) + 16 t^{6}$

Multiply

6

$6$ by

16

$16$ .

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 240 (r^{4} t^{2}) + 320 (r^{3} t^{3}) + 240 (r^{2} t^{4}) + 96 (r t^{5}) + 16 t^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 240 (r^{4} t^{2}) + 320 (r^{3} t^{3}) + 240 (r^{2} t^{4}) + 96 (r t^{5}) + 16 t^{6}$

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 240 (r^{4} t^{2}) + 320 (r^{3} t^{3}) + 240 (r^{2} t^{4}) + 96 (r t^{5}) + 16 t^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 (r^{5} t) + 240 (r^{4} t^{2}) + 320 (r^{3} t^{3}) + 240 (r^{2} t^{4}) + 96 (r t^{5}) + 16 t^{6}$

Remove parentheses.

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3}, 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$

Step 2

Factor out the GCF of

4

$4$ from

20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3} + 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$20 r^{3} + 60 r^{2} t + 60 r t^{2} + 20 t^{3} + 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$ .

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Factor out the GCF of

4

$4$ from each term in the polynomial.

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Factor out the GCF of

4

$4$ from the expression

20 r^{3}

$20 r^{3}$ .

4 (5 r^{3}) + 60 r^{2} t + 60 r t^{2} + 20 t^{3} + 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$4 (5 r^{3}) + 60 r^{2} t + 60 r t^{2} + 20 t^{3} + 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$

Factor out the GCF of

4

$4$ from the expression

60 r^{2} t

$60 r^{2} t$ .

4 (5 r^{3}) + 4 (15 r^{2} t) + 60 r t^{2} + 20 t^{3} + 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$4 (5 r^{3}) + 4 (15 r^{2} t) + 60 r t^{2} + 20 t^{3} + 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$

Factor out the GCF of

4

$4$ from the expression

60 r t^{2}

$60 r t^{2}$ .

4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 20 t^{3} + 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 20 t^{3} + 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$

Factor out the GCF of

4

$4$ from the expression

20 t^{3}

$20 t^{3}$ .

4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 16 r^{6} + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$

Factor out the GCF of

4

$4$ from the expression

16 r^{6}

$16 r^{6}$ .

4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 96 r^{5} t + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$

Factor out the GCF of

4

$4$ from the expression

96 r^{5} t

$96 r^{5} t$ .

4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 240 r^{4} t^{2} + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$

Factor out the GCF of

4

$4$ from the expression

240 r^{4} t^{2}

$240 r^{4} t^{2}$ .

4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 320 r^{3} t^{3} + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$

Factor out the GCF of

4

$4$ from the expression

320 r^{3} t^{3}

$320 r^{3} t^{3}$ .

4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 4 (80 r^{3} t^{3}) + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}

$4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 4 (80 r^{3} t^{3}) + 240 r^{2} t^{4} + 96 r t^{5} + 16 t^{6}$

Factor out the GCF of

4

$4$ from the expression

240 r^{2} t^{4}

$240 r^{2} t^{4}$ .

4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 4 (80 r^{3} t^{3}) + 4 (60 r^{2} t^{4}) + 96 r t^{5} + 16 t^{6}

$4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 4 (80 r^{3} t^{3}) + 4 (60 r^{2} t^{4}) + 96 r t^{5} + 16 t^{6}$

Factor out the GCF of

4

$4$ from the expression

96 r t^{5}

$96 r t^{5}$ .

4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 4 (80 r^{3} t^{3}) + 4 (60 r^{2} t^{4}) + 4 (24 r t^{5}) + 16 t^{6}

$4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 4 (80 r^{3} t^{3}) + 4 (60 r^{2} t^{4}) + 4 (24 r t^{5}) + 16 t^{6}$

Factor out the GCF of

4

$4$ from the expression

16 t^{6}

$16 t^{6}$ .

4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 4 (80 r^{3} t^{3}) + 4 (60 r^{2} t^{4}) + 4 (24 r t^{5}) + 4 (4 t^{6})

$4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 4 (80 r^{3} t^{3}) + 4 (60 r^{2} t^{4}) + 4 (24 r t^{5}) + 4 (4 t^{6})$

4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 4 (80 r^{3} t^{3}) + 4 (60 r^{2} t^{4}) + 4 (24 r t^{5}) + 4 (4 t^{6})

$4 (5 r^{3}) + 4 (15 r^{2} t) + 4 (15 r t^{2}) + 4 (5 t^{3}) + 4 (4 r^{6}) + 4 (24 r^{5} t) + 4 (60 r^{4} t^{2}) + 4 (80 r^{3} t^{3}) + 4 (60 r^{2} t^{4}) + 4 (24 r t^{5}) + 4 (4 t^{6})$

Since all the terms share a common factor of

4

$4$ , it can be factored out of each term.

4 (5 r^{3} + 15 r^{2} t + 15 r t^{2} + 5 t^{3} + 4 r^{6} + 24 r^{5} t + 60 r^{4} t^{2} + 80 r^{3} t^{3} + 60 r^{2} t^{4} + 24 r t^{5} + 4 t^{6})

$4 (5 r^{3} + 15 r^{2} t + 15 r t^{2} + 5 t^{3} + 4 r^{6} + 24 r^{5} t + 60 r^{4} t^{2} + 80 r^{3} t^{3} + 60 r^{2} t^{4} + 24 r t^{5} + 4 t^{6})$

4 (5 r^{3} + 15 r^{2} t + 15 r t^{2} + 5 t^{3} + 4 r^{6} + 24 r^{5} t + 60 r^{4} t^{2} + 80 r^{3} t^{3} + 60 r^{2} t^{4} + 24 r t^{5} + 4 t^{6})

$4 (5 r^{3} + 15 r^{2} t + 15 r t^{2} + 5 t^{3} + 4 r^{6} + 24 r^{5} t + 60 r^{4} t^{2} + 80 r^{3} t^{3} + 60 r^{2} t^{4} + 24 r t^{5} + 4 t^{6})$

Step 3

The greatest common factor

GCF

$GCF$ is the term in front of the factored expression.

4

$4$