Algebra Examples

$p (x) = (2 x + 2) (5 x - 2) (x + 3)$

Step 1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 1.1

Expand $(2 x + 2) (5 x - 2)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$p (x) = (2 x (5 x - 2) + 2 (5 x - 2)) (x + 3)$

Step 1.1.2

Apply the distributive property.

$p (x) = (2 x (5 x) + 2 x \cdot - 2 + 2 (5 x - 2)) (x + 3)$

Step 1.1.3

Apply the distributive property.

$p (x) = (2 x (5 x) + 2 x \cdot - 2 + 2 (5 x) + 2 \cdot - 2) (x + 3)$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$p (x) = (2 \cdot (5 x \cdot x) + 2 x \cdot - 2 + 2 (5 x) + 2 \cdot - 2) (x + 3)$

Step 1.2.1.2

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

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Step 1.2.1.2.1

Move $x$ .

$p (x) = (2 \cdot (5 (x \cdot x)) + 2 x \cdot - 2 + 2 (5 x) + 2 \cdot - 2) (x + 3)$

Step 1.2.1.2.2

Multiply $x$ by $x$ .

$p (x) = (2 \cdot (5 x^{2}) + 2 x \cdot - 2 + 2 (5 x) + 2 \cdot - 2) (x + 3)$

Step 1.2.1.3

Multiply $2$ by $5$ .

$p (x) = (10 x^{2} + 2 x \cdot - 2 + 2 (5 x) + 2 \cdot - 2) (x + 3)$

Step 1.2.1.4

Multiply $- 2$ by $2$ .

$p (x) = (10 x^{2} - 4 x + 2 (5 x) + 2 \cdot - 2) (x + 3)$

Step 1.2.1.5

Multiply $5$ by $2$ .

$p (x) = (10 x^{2} - 4 x + 10 x + 2 \cdot - 2) (x + 3)$

Step 1.2.1.6

Multiply $2$ by $- 2$ .

$p (x) = (10 x^{2} - 4 x + 10 x - 4) (x + 3)$

Step 1.2.2

Add $- 4 x$ and $10 x$ .

$p (x) = (10 x^{2} + 6 x - 4) (x + 3)$

Step 1.3

Expand $(10 x^{2} + 6 x - 4) (x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$p (x) = 10 x^{2} x + 10 x^{2} \cdot 3 + 6 x \cdot x + 6 x \cdot 3 - 4 x - 4 \cdot 3$

Step 1.4

Simplify terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

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

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Step 1.4.1.1.1

Move $x$ .

$p (x) = 10 (x \cdot x^{2}) + 10 x^{2} \cdot 3 + 6 x \cdot x + 6 x \cdot 3 - 4 x - 4 \cdot 3$

Step 1.4.1.1.2

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

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Step 1.4.1.1.2.1

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

$p (x) = 10 (x \cdot x^{2}) + 10 x^{2} \cdot 3 + 6 x \cdot x + 6 x \cdot 3 - 4 x - 4 \cdot 3$

Step 1.4.1.1.2.2

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

$p (x) = 10 x^{1 + 2} + 10 x^{2} \cdot 3 + 6 x \cdot x + 6 x \cdot 3 - 4 x - 4 \cdot 3$

Step 1.4.1.1.3

Add $1$ and $2$ .

$p (x) = 10 x^{3} + 10 x^{2} \cdot 3 + 6 x \cdot x + 6 x \cdot 3 - 4 x - 4 \cdot 3$

Step 1.4.1.2

Multiply $3$ by $10$ .

$p (x) = 10 x^{3} + 30 x^{2} + 6 x \cdot x + 6 x \cdot 3 - 4 x - 4 \cdot 3$

Step 1.4.1.3

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

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Step 1.4.1.3.1

Move $x$ .

$p (x) = 10 x^{3} + 30 x^{2} + 6 (x \cdot x) + 6 x \cdot 3 - 4 x - 4 \cdot 3$

Step 1.4.1.3.2

Multiply $x$ by $x$ .

$p (x) = 10 x^{3} + 30 x^{2} + 6 x^{2} + 6 x \cdot 3 - 4 x - 4 \cdot 3$

Step 1.4.1.4

Multiply $3$ by $6$ .

$p (x) = 10 x^{3} + 30 x^{2} + 6 x^{2} + 18 x - 4 x - 4 \cdot 3$

Step 1.4.1.5

Multiply $- 4$ by $3$ .

$p (x) = 10 x^{3} + 30 x^{2} + 6 x^{2} + 18 x - 4 x - 12$

Step 1.4.2

Simplify by adding terms.

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Step 1.4.2.1

Add $30 x^{2}$ and $6 x^{2}$ .

$p (x) = 10 x^{3} + 36 x^{2} + 18 x - 4 x - 12$

Step 1.4.2.2

Subtract $4 x$ from $18 x$ .

$p (x) = 10 x^{3} + 36 x^{2} + 14 x - 12$

Step 2

The degree of a polynomial is the highest degree of its terms.

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Step 2.1

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$10 x^{3} \to 3$

$36 x^{2} \to 2$

$14 x \to 1$

$- 12 \to 0$

Step 2.2

The largest exponent is the degree of the polynomial.

$3$

Step 3

The leading term in a polynomial is the term with the highest degree.

$10 x^{3}$

Step 4

The leading coefficient of a polynomial is the coefficient of the leading term.

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Step 4.1

The leading term in a polynomial is the term with the highest degree.

$10 x^{3}$

Step 4.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$10$

Step 5

List the results.

Polynomial Degree: $3$

Leading Term: $10 x^{3}$

Leading Coefficient: $10$