Algebra Examples

${(x + 5)}^{4} + 2 {(x + 5)}^{2} - 1$

Step 1

Use the Binomial Theorem.

$x^{4} + 4 x^{3} \cdot 5 + 6 x^{2} \cdot 5^{2} + 4 x \cdot 5^{3} + 5^{4} + 2 {(x + 5)}^{2} - 1$

Step 2

Simplify each term.

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Step 2.1

Multiply $5$ by $4$ .

$x^{4} + 20 x^{3} + 6 x^{2} \cdot 5^{2} + 4 x \cdot 5^{3} + 5^{4} + 2 {(x + 5)}^{2} - 1$

Step 2.2

Raise $5$ to the power of $2$ .

$x^{4} + 20 x^{3} + 6 x^{2} \cdot 25 + 4 x \cdot 5^{3} + 5^{4} + 2 {(x + 5)}^{2} - 1$

Step 2.3

Multiply $25$ by $6$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 4 x \cdot 5^{3} + 5^{4} + 2 {(x + 5)}^{2} - 1$

Step 2.4

Raise $5$ to the power of $3$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 4 x \cdot 125 + 5^{4} + 2 {(x + 5)}^{2} - 1$

Step 2.5

Multiply $125$ by $4$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 5^{4} + 2 {(x + 5)}^{2} - 1$

Step 2.6

Raise $5$ to the power of $4$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 {(x + 5)}^{2} - 1$

Step 3

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 ((x + 5) (x + 5)) - 1$

Step 4

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 (x (x + 5) + 5 (x + 5)) - 1$

Step 4.2

Apply the distributive property.

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 (x \cdot x + x \cdot 5 + 5 (x + 5)) - 1$

Step 4.3

Apply the distributive property.

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) - 1$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $x$ by $x$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) - 1$

Step 5.1.2

Move $5$ to the left of $x$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 (x^{2} + 5 x + 5 x + 5 \cdot 5) - 1$

Step 5.1.3

Multiply $5$ by $5$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 (x^{2} + 5 x + 5 x + 25) - 1$

Step 5.2

Add $5 x$ and $5 x$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 (x^{2} + 10 x + 25) - 1$

Step 6

Apply the distributive property.

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 x^{2} + 2 (10 x) + 2 \cdot 25 - 1$

Step 7

Simplify.

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Step 7.1

Multiply $10$ by $2$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 x^{2} + 20 x + 2 \cdot 25 - 1$

Step 7.2

Multiply $2$ by $25$ .

$x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625 + 2 x^{2} + 20 x + 50 - 1$

Step 8

Add $150 x^{2}$ and $2 x^{2}$ .

$x^{4} + 20 x^{3} + 152 x^{2} + 500 x + 625 + 20 x + 50 - 1$

Step 9

Add $500 x$ and $20 x$ .

$x^{4} + 20 x^{3} + 152 x^{2} + 520 x + 625 + 50 - 1$

Step 10

Add $625$ and $50$ .

$x^{4} + 20 x^{3} + 152 x^{2} + 520 x + 675 - 1$

Step 11

Subtract $1$ from $675$ .

$x^{4} + 20 x^{3} + 152 x^{2} + 520 x + 674$