Algebra Examples

$f (t) = 3 t (t - 3) (t + 4)$

Step 1

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

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Step 1.1

Apply the distributive property.

$f (t) = 3 t (t - 3) t + 3 t (t - 3) \cdot 4$

Step 1.2

Simplify the expression.

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Step 1.2.1

Multiply $4$ by $3$ .

$f (t) = 3 t (t - 3) t + 12 t (t - 3)$

Step 1.2.2

Reorder factors in $3 t (t - 3) t + 12 t (t - 3)$ .

$f (t) = 3 t \cdot t (t - 3) + 12 t (t - 3)$

Step 2

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

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Step 2.1

Simplify and reorder the polynomial.

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Multiply $t$ by $t$ by adding the exponents.

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Step 2.1.1.1.1

Move $t$ .

$3 (t \cdot t) (t - 3) + 12 t (t - 3)$

Step 2.1.1.1.2

Multiply $t$ by $t$ .

$3 t^{2} (t - 3) + 12 t (t - 3)$

Step 2.1.1.2

Apply the distributive property.

$3 t^{2} t + 3 t^{2} \cdot - 3 + 12 t (t - 3)$

Step 2.1.1.3

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

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Step 2.1.1.3.1

Move $t$ .

$3 (t \cdot t^{2}) + 3 t^{2} \cdot - 3 + 12 t (t - 3)$

Step 2.1.1.3.2

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

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Step 2.1.1.3.2.1

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

$3 (t^{1} t^{2}) + 3 t^{2} \cdot - 3 + 12 t (t - 3)$

Step 2.1.1.3.2.2

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

$3 t^{1 + 2} + 3 t^{2} \cdot - 3 + 12 t (t - 3)$

Step 2.1.1.3.3

Add $1$ and $2$ .

$3 t^{3} + 3 t^{2} \cdot - 3 + 12 t (t - 3)$

Step 2.1.1.4

Multiply $- 3$ by $3$ .

$3 t^{3} - 9 t^{2} + 12 t (t - 3)$

Step 2.1.1.5

Apply the distributive property.

$3 t^{3} - 9 t^{2} + 12 t \cdot t + 12 t \cdot - 3$

Step 2.1.1.6

Multiply $t$ by $t$ by adding the exponents.

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Step 2.1.1.6.1

Move $t$ .

$3 t^{3} - 9 t^{2} + 12 (t \cdot t) + 12 t \cdot - 3$

Step 2.1.1.6.2

Multiply $t$ by $t$ .

$3 t^{3} - 9 t^{2} + 12 t^{2} + 12 t \cdot - 3$

Step 2.1.1.7

Multiply $- 3$ by $12$ .

$3 t^{3} - 9 t^{2} + 12 t^{2} - 36 t$

Step 2.1.2

Add $- 9 t^{2}$ and $12 t^{2}$ .

$3 t^{3} + 3 t^{2} - 36 t$

Step 2.2

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

$3 t^{3} \to 3$

$3 t^{2} \to 2$

$- 36 t \to 1$

Step 2.3

The largest exponent is the degree of the polynomial.

$3$

Step 3

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

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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

Simplify each term.

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Step 3.1.1.1

Multiply $t$ by $t$ by adding the exponents.

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Step 3.1.1.1.1

Move $t$ .

$3 (t \cdot t) (t - 3) + 12 t (t - 3)$

Step 3.1.1.1.2

Multiply $t$ by $t$ .

$3 t^{2} (t - 3) + 12 t (t - 3)$

Step 3.1.1.2

Apply the distributive property.

$3 t^{2} t + 3 t^{2} \cdot - 3 + 12 t (t - 3)$

Step 3.1.1.3

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

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Step 3.1.1.3.1

Move $t$ .

$3 (t \cdot t^{2}) + 3 t^{2} \cdot - 3 + 12 t (t - 3)$

Step 3.1.1.3.2

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

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Step 3.1.1.3.2.1

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

$3 (t^{1} t^{2}) + 3 t^{2} \cdot - 3 + 12 t (t - 3)$

Step 3.1.1.3.2.2

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

$3 t^{1 + 2} + 3 t^{2} \cdot - 3 + 12 t (t - 3)$

Step 3.1.1.3.3

Add $1$ and $2$ .

$3 t^{3} + 3 t^{2} \cdot - 3 + 12 t (t - 3)$

Step 3.1.1.4

Multiply $- 3$ by $3$ .

$3 t^{3} - 9 t^{2} + 12 t (t - 3)$

Step 3.1.1.5

Apply the distributive property.

$3 t^{3} - 9 t^{2} + 12 t \cdot t + 12 t \cdot - 3$

Step 3.1.1.6

Multiply $t$ by $t$ by adding the exponents.

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Step 3.1.1.6.1

Move $t$ .

$3 t^{3} - 9 t^{2} + 12 (t \cdot t) + 12 t \cdot - 3$

Step 3.1.1.6.2

Multiply $t$ by $t$ .

$3 t^{3} - 9 t^{2} + 12 t^{2} + 12 t \cdot - 3$

Step 3.1.1.7

Multiply $- 3$ by $12$ .

$3 t^{3} - 9 t^{2} + 12 t^{2} - 36 t$

Step 3.1.2

Add $- 9 t^{2}$ and $12 t^{2}$ .

$3 t^{3} + 3 t^{2} - 36 t$

Step 3.2

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

$3 t^{3}$

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.

$3 t^{3}$

Step 4.2

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

$3$

Step 5

List the results.

Polynomial Degree: $3$

Leading Term: $3 t^{3}$

Leading Coefficient: $3$