Algebra Examples

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

Step 1

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

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Step 1.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$f (x) = x^{3} ((x + 3) (x + 3)) (x - 5)$

Step 1.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = x^{3} (x (x + 3) + 3 (x + 3)) (x - 5)$

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Move $3$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $3$ by $3$ .

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

Step 1.3.2

Add $3 x$ and $3 x$ .

$f (x) = x^{3} (x^{2} + 6 x + 9) (x - 5)$

Step 1.4

Apply the distributive property.

$f (x) = (x^{3} x^{2} + x^{3} (6 x) + x^{3} \cdot 9) (x - 5)$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 1.5.1.1

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

$f (x) = (x^{3 + 2} + x^{3} (6 x) + x^{3} \cdot 9) (x - 5)$

Step 1.5.1.2

Add $3$ and $2$ .

$f (x) = (x^{5} + x^{3} (6 x) + x^{3} \cdot 9) (x - 5)$

Step 1.5.2

Rewrite using the commutative property of multiplication.

$f (x) = (x^{5} + 6 x^{3} x + x^{3} \cdot 9) (x - 5)$

Step 1.5.3

Move $9$ to the left of $x^{3}$ .

$f (x) = (x^{5} + 6 x^{3} x + 9 \cdot x^{3}) (x - 5)$

Step 1.6

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.6.1

Move $x$ .

$f (x) = (x^{5} + 6 (x \cdot x^{3}) + 9 \cdot x^{3}) (x - 5)$

Step 1.6.2

Multiply $x$ by $x^{3}$ .

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Step 1.6.2.1

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

$f (x) = (x^{5} + 6 (x \cdot x^{3}) + 9 \cdot x^{3}) (x - 5)$

Step 1.6.2.2

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

$f (x) = (x^{5} + 6 x^{1 + 3} + 9 \cdot x^{3}) (x - 5)$

Step 1.6.3

Add $1$ and $3$ .

$f (x) = (x^{5} + 6 x^{4} + 9 \cdot x^{3}) (x - 5)$

$f (x) = (x^{5} + 6 x^{4} + 9 x^{3}) (x - 5)$

Step 1.7

Expand $(x^{5} + 6 x^{4} + 9 x^{3}) (x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = x^{5} x + x^{5} \cdot - 5 + 6 x^{4} x + 6 x^{4} \cdot - 5 + 9 x^{3} x + 9 x^{3} \cdot - 5$

Step 1.8

Simplify terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 1.8.1.1.1

Multiply $x^{5}$ by $x$ .

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Step 1.8.1.1.1.1

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

$f (x) = x^{5} x + x^{5} \cdot - 5 + 6 x^{4} x + 6 x^{4} \cdot - 5 + 9 x^{3} x + 9 x^{3} \cdot - 5$

Step 1.8.1.1.1.2

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

$f (x) = x^{5 + 1} + x^{5} \cdot - 5 + 6 x^{4} x + 6 x^{4} \cdot - 5 + 9 x^{3} x + 9 x^{3} \cdot - 5$

Step 1.8.1.1.2

Add $5$ and $1$ .

$f (x) = x^{6} + x^{5} \cdot - 5 + 6 x^{4} x + 6 x^{4} \cdot - 5 + 9 x^{3} x + 9 x^{3} \cdot - 5$

Step 1.8.1.2

Move $- 5$ to the left of $x^{5}$ .

$f (x) = x^{6} - 5 \cdot x^{5} + 6 x^{4} x + 6 x^{4} \cdot - 5 + 9 x^{3} x + 9 x^{3} \cdot - 5$

Step 1.8.1.3

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 1.8.1.3.1

Move $x$ .

$f (x) = x^{6} - 5 x^{5} + 6 (x \cdot x^{4}) + 6 x^{4} \cdot - 5 + 9 x^{3} x + 9 x^{3} \cdot - 5$

Step 1.8.1.3.2

Multiply $x$ by $x^{4}$ .

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Step 1.8.1.3.2.1

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

$f (x) = x^{6} - 5 x^{5} + 6 (x \cdot x^{4}) + 6 x^{4} \cdot - 5 + 9 x^{3} x + 9 x^{3} \cdot - 5$

Step 1.8.1.3.2.2

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

$f (x) = x^{6} - 5 x^{5} + 6 x^{1 + 4} + 6 x^{4} \cdot - 5 + 9 x^{3} x + 9 x^{3} \cdot - 5$

Step 1.8.1.3.3

Add $1$ and $4$ .

$f (x) = x^{6} - 5 x^{5} + 6 x^{5} + 6 x^{4} \cdot - 5 + 9 x^{3} x + 9 x^{3} \cdot - 5$

Step 1.8.1.4

Multiply $- 5$ by $6$ .

$f (x) = x^{6} - 5 x^{5} + 6 x^{5} - 30 x^{4} + 9 x^{3} x + 9 x^{3} \cdot - 5$

Step 1.8.1.5

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.8.1.5.1

Move $x$ .

$f (x) = x^{6} - 5 x^{5} + 6 x^{5} - 30 x^{4} + 9 (x \cdot x^{3}) + 9 x^{3} \cdot - 5$

Step 1.8.1.5.2

Multiply $x$ by $x^{3}$ .

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Step 1.8.1.5.2.1

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

$f (x) = x^{6} - 5 x^{5} + 6 x^{5} - 30 x^{4} + 9 (x \cdot x^{3}) + 9 x^{3} \cdot - 5$

Step 1.8.1.5.2.2

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

$f (x) = x^{6} - 5 x^{5} + 6 x^{5} - 30 x^{4} + 9 x^{1 + 3} + 9 x^{3} \cdot - 5$

Step 1.8.1.5.3

Add $1$ and $3$ .

$f (x) = x^{6} - 5 x^{5} + 6 x^{5} - 30 x^{4} + 9 x^{4} + 9 x^{3} \cdot - 5$

Step 1.8.1.6

Multiply $- 5$ by $9$ .

$f (x) = x^{6} - 5 x^{5} + 6 x^{5} - 30 x^{4} + 9 x^{4} - 45 x^{3}$

Step 1.8.2

Simplify by adding terms.

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Step 1.8.2.1

Add $- 5 x^{5}$ and $6 x^{5}$ .

$f (x) = x^{6} + x^{5} - 30 x^{4} + 9 x^{4} - 45 x^{3}$

Step 1.8.2.2

Add $- 30 x^{4}$ and $9 x^{4}$ .

$f (x) = x^{6} + x^{5} - 21 x^{4} - 45 x^{3}$

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.

$x^{6} \to 6$

$x^{5} \to 5$

$- 21 x^{4} \to 4$

$- 45 x^{3} \to 3$

Step 2.2

The largest exponent is the degree of the polynomial.

$6$

Step 3

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

$x^{6}$

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.

$x^{6}$

Step 4.2

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

$1$

Step 5

List the results.

Polynomial Degree: $6$

Leading Term: $x^{6}$

Leading Coefficient: $1$