Algebra Examples

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

Step 1

Identify the degree of the function.

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Step 1.1

Simplify and reorder the polynomial.

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Step 1.1.1

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

$x ((x + 5) (x + 5)) (x + 3)$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

$x (x (x + 5) + 5 (x + 5)) (x + 3)$

Step 1.1.2.2

Apply the distributive property.

$x (x \cdot x + x \cdot 5 + 5 (x + 5)) (x + 3)$

Step 1.1.2.3

Apply the distributive property.

$x (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) (x + 3)$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

Multiply $5$ by $5$ .

$x (x^{2} + 5 x + 5 x + 25) (x + 3)$

Step 1.1.3.2

Add $5 x$ and $5 x$ .

$x (x^{2} + 10 x + 25) (x + 3)$

Step 1.1.4

Apply the distributive property.

$(x \cdot x^{2} + x (10 x) + x \cdot 25) (x + 3)$

Step 1.1.5

Simplify.

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Step 1.1.5.1

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

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Step 1.1.5.1.1

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

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Step 1.1.5.1.1.1

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

$(x^{1} x^{2} + x (10 x) + x \cdot 25) (x + 3)$

Step 1.1.5.1.1.2

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

$(x^{1 + 2} + x (10 x) + x \cdot 25) (x + 3)$

Step 1.1.5.1.2

Add $1$ and $2$ .

$(x^{3} + x (10 x) + x \cdot 25) (x + 3)$

Step 1.1.5.2

Rewrite using the commutative property of multiplication.

$(x^{3} + 10 x \cdot x + x \cdot 25) (x + 3)$

Step 1.1.5.3

Move $25$ to the left of $x$ .

$(x^{3} + 10 x \cdot x + 25 \cdot x) (x + 3)$

Step 1.1.6

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

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Step 1.1.6.1

Move $x$ .

$(x^{3} + 10 (x \cdot x) + 25 \cdot x) (x + 3)$

Step 1.1.6.2

Multiply $x$ by $x$ .

$(x^{3} + 10 x^{2} + 25 \cdot x) (x + 3)$

$(x^{3} + 10 x^{2} + 25 x) (x + 3)$

Step 1.1.7

Expand $(x^{3} + 10 x^{2} + 25 x) (x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x + x^{3} \cdot 3 + 10 x^{2} x + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 1.1.8

Simplify terms.

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

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

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Step 1.1.8.1.1.1

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

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Step 1.1.8.1.1.1.1

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

$x^{3} x^{1} + x^{3} \cdot 3 + 10 x^{2} x + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 1.1.8.1.1.1.2

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

$x^{3 + 1} + x^{3} \cdot 3 + 10 x^{2} x + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 1.1.8.1.1.2

Add $3$ and $1$ .

$x^{4} + x^{3} \cdot 3 + 10 x^{2} x + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 1.1.8.1.2

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

$x^{4} + 3 \cdot x^{3} + 10 x^{2} x + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 1.1.8.1.3

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

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Step 1.1.8.1.3.1

Move $x$ .

$x^{4} + 3 x^{3} + 10 (x \cdot x^{2}) + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 1.1.8.1.3.2

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

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Step 1.1.8.1.3.2.1

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

$x^{4} + 3 x^{3} + 10 (x^{1} x^{2}) + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 1.1.8.1.3.2.2

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

$x^{4} + 3 x^{3} + 10 x^{1 + 2} + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 1.1.8.1.3.3

Add $1$ and $2$ .

$x^{4} + 3 x^{3} + 10 x^{3} + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 1.1.8.1.4

Multiply $3$ by $10$ .

$x^{4} + 3 x^{3} + 10 x^{3} + 30 x^{2} + 25 x \cdot x + 25 x \cdot 3$

Step 1.1.8.1.5

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

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Step 1.1.8.1.5.1

Move $x$ .

$x^{4} + 3 x^{3} + 10 x^{3} + 30 x^{2} + 25 (x \cdot x) + 25 x \cdot 3$

Step 1.1.8.1.5.2

Multiply $x$ by $x$ .

$x^{4} + 3 x^{3} + 10 x^{3} + 30 x^{2} + 25 x^{2} + 25 x \cdot 3$

Step 1.1.8.1.6

Multiply $3$ by $25$ .

$x^{4} + 3 x^{3} + 10 x^{3} + 30 x^{2} + 25 x^{2} + 75 x$

Step 1.1.8.2

Simplify by adding terms.

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Step 1.1.8.2.1

Add $3 x^{3}$ and $10 x^{3}$ .

$x^{4} + 13 x^{3} + 30 x^{2} + 25 x^{2} + 75 x$

Step 1.1.8.2.2

Add $30 x^{2}$ and $25 x^{2}$ .

$x^{4} + 13 x^{3} + 55 x^{2} + 75 x$

Step 1.2

The largest exponent is the degree of the polynomial.

$4$

Step 2

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 3

Identify the leading coefficient.

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Step 3.1

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

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Step 3.1.1

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

$x ((x + 5) (x + 5)) (x + 3)$

Step 3.1.2

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

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Step 3.1.2.1

Apply the distributive property.

$x (x (x + 5) + 5 (x + 5)) (x + 3)$

Step 3.1.2.2

Apply the distributive property.

$x (x \cdot x + x \cdot 5 + 5 (x + 5)) (x + 3)$

Step 3.1.2.3

Apply the distributive property.

$x (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) (x + 3)$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.3.1.2

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

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

Step 3.1.3.1.3

Multiply $5$ by $5$ .

$x (x^{2} + 5 x + 5 x + 25) (x + 3)$

Step 3.1.3.2

Add $5 x$ and $5 x$ .

$x (x^{2} + 10 x + 25) (x + 3)$

Step 3.1.4

Apply the distributive property.

$(x \cdot x^{2} + x (10 x) + x \cdot 25) (x + 3)$

Step 3.1.5

Simplify.

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Step 3.1.5.1

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

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Step 3.1.5.1.1

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

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Step 3.1.5.1.1.1

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

$(x^{1} x^{2} + x (10 x) + x \cdot 25) (x + 3)$

Step 3.1.5.1.1.2

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

$(x^{1 + 2} + x (10 x) + x \cdot 25) (x + 3)$

Step 3.1.5.1.2

Add $1$ and $2$ .

$(x^{3} + x (10 x) + x \cdot 25) (x + 3)$

Step 3.1.5.2

Rewrite using the commutative property of multiplication.

$(x^{3} + 10 x \cdot x + x \cdot 25) (x + 3)$

Step 3.1.5.3

Move $25$ to the left of $x$ .

$(x^{3} + 10 x \cdot x + 25 \cdot x) (x + 3)$

Step 3.1.6

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

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Step 3.1.6.1

Move $x$ .

$(x^{3} + 10 (x \cdot x) + 25 \cdot x) (x + 3)$

Step 3.1.6.2

Multiply $x$ by $x$ .

$(x^{3} + 10 x^{2} + 25 \cdot x) (x + 3)$

$(x^{3} + 10 x^{2} + 25 x) (x + 3)$

Step 3.1.7

Expand $(x^{3} + 10 x^{2} + 25 x) (x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x + x^{3} \cdot 3 + 10 x^{2} x + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 3.1.8

Simplify terms.

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Step 3.1.8.1

Simplify each term.

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Step 3.1.8.1.1

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

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Step 3.1.8.1.1.1

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

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Step 3.1.8.1.1.1.1

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

$x^{3} x^{1} + x^{3} \cdot 3 + 10 x^{2} x + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 3.1.8.1.1.1.2

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

$x^{3 + 1} + x^{3} \cdot 3 + 10 x^{2} x + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 3.1.8.1.1.2

Add $3$ and $1$ .

$x^{4} + x^{3} \cdot 3 + 10 x^{2} x + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 3.1.8.1.2

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

$x^{4} + 3 \cdot x^{3} + 10 x^{2} x + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 3.1.8.1.3

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

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Step 3.1.8.1.3.1

Move $x$ .

$x^{4} + 3 x^{3} + 10 (x \cdot x^{2}) + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 3.1.8.1.3.2

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

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Step 3.1.8.1.3.2.1

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

$x^{4} + 3 x^{3} + 10 (x^{1} x^{2}) + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 3.1.8.1.3.2.2

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

$x^{4} + 3 x^{3} + 10 x^{1 + 2} + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 3.1.8.1.3.3

Add $1$ and $2$ .

$x^{4} + 3 x^{3} + 10 x^{3} + 10 x^{2} \cdot 3 + 25 x \cdot x + 25 x \cdot 3$

Step 3.1.8.1.4

Multiply $3$ by $10$ .

$x^{4} + 3 x^{3} + 10 x^{3} + 30 x^{2} + 25 x \cdot x + 25 x \cdot 3$

Step 3.1.8.1.5

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

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Step 3.1.8.1.5.1

Move $x$ .

$x^{4} + 3 x^{3} + 10 x^{3} + 30 x^{2} + 25 (x \cdot x) + 25 x \cdot 3$

Step 3.1.8.1.5.2

Multiply $x$ by $x$ .

$x^{4} + 3 x^{3} + 10 x^{3} + 30 x^{2} + 25 x^{2} + 25 x \cdot 3$

Step 3.1.8.1.6

Multiply $3$ by $25$ .

$x^{4} + 3 x^{3} + 10 x^{3} + 30 x^{2} + 25 x^{2} + 75 x$

Step 3.1.8.2

Simplify by adding terms.

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Step 3.1.8.2.1

Add $3 x^{3}$ and $10 x^{3}$ .

$x^{4} + 13 x^{3} + 30 x^{2} + 25 x^{2} + 75 x$

Step 3.1.8.2.2

Add $30 x^{2}$ and $25 x^{2}$ .

$x^{4} + 13 x^{3} + 55 x^{2} + 75 x$

Step 3.2

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

$x^{4}$

Step 3.3

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

$1$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Rises to the left and rises to the right

Step 7