Find the Degree, Leading Term, and Leading Coefficient h(x)=3x^2(x-2)(x-6)

$h (x) = 3 x^{2} (x - 2) (x - 6)$

Step 1

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

Apply the distributive property.

$h (x) = (3 x^{2} x + 3 x^{2} \cdot - 2) (x - 6)$

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

Move $x$ .

$h (x) = (3 (x \cdot x^{2}) + 3 x^{2} \cdot - 2) (x - 6)$

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

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

$h (x) = (3 (x \cdot x^{2}) + 3 x^{2} \cdot - 2) (x - 6)$

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

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

Add $1$ and $2$ .

$h (x) = (3 x^{3} + 3 x^{2} \cdot - 2) (x - 6)$

Multiply $- 2$ by $3$ .

$h (x) = (3 x^{3} - 6 x^{2}) (x - 6)$

Expand $(3 x^{3} - 6 x^{2}) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$h (x) = 3 x^{3} (x - 6) - 6 x^{2} (x - 6)$

Apply the distributive property.

$h (x) = 3 x^{3} x + 3 x^{3} \cdot - 6 - 6 x^{2} (x - 6)$

Apply the distributive property.

$h (x) = 3 x^{3} x + 3 x^{3} \cdot - 6 - 6 x^{2} x - 6 x^{2} \cdot - 6$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$h (x) = 3 (x \cdot x^{3}) + 3 x^{3} \cdot - 6 - 6 x^{2} x - 6 x^{2} \cdot - 6$

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

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

$h (x) = 3 (x \cdot x^{3}) + 3 x^{3} \cdot - 6 - 6 x^{2} x - 6 x^{2} \cdot - 6$

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

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

Add $1$ and $3$ .

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

Multiply $- 6$ by $3$ .

$h (x) = 3 x^{4} - 18 x^{3} - 6 x^{2} x - 6 x^{2} \cdot - 6$

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

Move $x$ .

$h (x) = 3 x^{4} - 18 x^{3} - 6 (x \cdot x^{2}) - 6 x^{2} \cdot - 6$

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

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

$h (x) = 3 x^{4} - 18 x^{3} - 6 (x \cdot x^{2}) - 6 x^{2} \cdot - 6$

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

$h (x) = 3 x^{4} - 18 x^{3} - 6 x^{1 + 2} - 6 x^{2} \cdot - 6$

Add $1$ and $2$ .

$h (x) = 3 x^{4} - 18 x^{3} - 6 x^{3} - 6 x^{2} \cdot - 6$

Multiply $- 6$ by $- 6$ .

$h (x) = 3 x^{4} - 18 x^{3} - 6 x^{3} + 36 x^{2}$

Subtract $6 x^{3}$ from $- 18 x^{3}$ .

$h (x) = 3 x^{4} - 24 x^{3} + 36 x^{2}$

Step 2

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

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

$3 x^{4} \to 4$

$- 24 x^{3} \to 3$

$36 x^{2} \to 2$

The largest exponent is the degree of the polynomial.

$4$

Step 3

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

$3 x^{4}$

Step 4

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

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

$3 x^{4}$

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

$3$

Step 5

List the results.

Polynomial Degree: $4$

Leading Term: $3 x^{4}$

Leading Coefficient: $3$

Find the Degree, Leading Term, and Leading Coefficient h(x)=3x^2(x-2)(x-6)

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