Algebra Examples

$f (x) = (5 x - 3) (25 x^{2} + 15 x + 9)$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$f (x) = a x^{2} + b x + c$

Step 2

Expand $(5 x - 3) (25 x^{2} + 15 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = 5 x (25 x^{2}) + 5 x (15 x) + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$f (x) = 5 \cdot (25 x \cdot x^{2}) + 5 x (15 x) + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.1.2.1

Move $x^{2}$ .

$f (x) = 5 \cdot (25 (x^{2} x)) + 5 x (15 x) + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.2.2

Multiply $x^{2}$ by $x$ .

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Step 3.1.2.2.1

Raise $x$ to the power of $1$ .

$f (x) = 5 \cdot (25 (x^{2} x)) + 5 x (15 x) + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (x) = 5 \cdot (25 x^{2 + 1}) + 5 x (15 x) + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.2.3

Add $2$ and $1$ .

$f (x) = 5 \cdot (25 x^{3}) + 5 x (15 x) + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.3

Multiply $5$ by $25$ .

$f (x) = 125 x^{3} + 5 x (15 x) + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$f (x) = 125 x^{3} + 5 \cdot (15 x \cdot x) + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.5.1

Move $x$ .

$f (x) = 125 x^{3} + 5 \cdot (15 (x \cdot x)) + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.5.2

Multiply $x$ by $x$ .

$f (x) = 125 x^{3} + 5 \cdot (15 x^{2}) + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.6

Multiply $5$ by $15$ .

$f (x) = 125 x^{3} + 75 x^{2} + 5 x \cdot 9 - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.7

Multiply $9$ by $5$ .

$f (x) = 125 x^{3} + 75 x^{2} + 45 x - 3 (25 x^{2}) - 3 (15 x) - 3 \cdot 9$

Step 3.1.8

Multiply $25$ by $- 3$ .

$f (x) = 125 x^{3} + 75 x^{2} + 45 x - 75 x^{2} - 3 (15 x) - 3 \cdot 9$

Step 3.1.9

Multiply $15$ by $- 3$ .

$f (x) = 125 x^{3} + 75 x^{2} + 45 x - 75 x^{2} - 45 x - 3 \cdot 9$

Step 3.1.10

Multiply $- 3$ by $9$ .

$f (x) = 125 x^{3} + 75 x^{2} + 45 x - 75 x^{2} - 45 x - 27$

Step 3.2

Combine the opposite terms in $125 x^{3} + 75 x^{2} + 45 x - 75 x^{2} - 45 x - 27$ .

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Step 3.2.1

Subtract $75 x^{2}$ from $75 x^{2}$ .

$f (x) = 125 x^{3} + 45 x + 0 - 45 x - 27$

Step 3.2.2

Add $125 x^{3} + 45 x$ and $0$ .

$f (x) = 125 x^{3} + 45 x - 45 x - 27$

Step 3.2.3

Subtract $45 x$ from $45 x$ .

$f (x) = 125 x^{3} + 0 - 27$

Step 3.2.4

Add $125 x^{3}$ and $0$ .

$f (x) = 125 x^{3} - 27$