Find the Degree, Leading Term, and Leading Coefficient g(x)=-7x^2(x-2)(x-3)

$g (x) = - 7 x^{2} (x - 2) (x - 3)$

Step 1

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

Apply the distributive property.

$g (x) = (- 7 x^{2} x - 7 x^{2} \cdot - 2) (x - 3)$

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

Move $x$ .

$g (x) = (- 7 (x \cdot x^{2}) - 7 x^{2} \cdot - 2) (x - 3)$

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

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

$g (x) = (- 7 (x \cdot x^{2}) - 7 x^{2} \cdot - 2) (x - 3)$

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

$g (x) = (- 7 x^{1 + 2} - 7 x^{2} \cdot - 2) (x - 3)$

Add $1$ and $2$ .

$g (x) = (- 7 x^{3} - 7 x^{2} \cdot - 2) (x - 3)$

Multiply $- 2$ by $- 7$ .

$g (x) = (- 7 x^{3} + 14 x^{2}) (x - 3)$

Expand $(- 7 x^{3} + 14 x^{2}) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$g (x) = - 7 x^{3} (x - 3) + 14 x^{2} (x - 3)$

Apply the distributive property.

$g (x) = - 7 x^{3} x - 7 x^{3} \cdot - 3 + 14 x^{2} (x - 3)$

Apply the distributive property.

$g (x) = - 7 x^{3} x - 7 x^{3} \cdot - 3 + 14 x^{2} x + 14 x^{2} \cdot - 3$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$g (x) = - 7 (x \cdot x^{3}) - 7 x^{3} \cdot - 3 + 14 x^{2} x + 14 x^{2} \cdot - 3$

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

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

$g (x) = - 7 (x \cdot x^{3}) - 7 x^{3} \cdot - 3 + 14 x^{2} x + 14 x^{2} \cdot - 3$

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

$g (x) = - 7 x^{1 + 3} - 7 x^{3} \cdot - 3 + 14 x^{2} x + 14 x^{2} \cdot - 3$

Add $1$ and $3$ .

$g (x) = - 7 x^{4} - 7 x^{3} \cdot - 3 + 14 x^{2} x + 14 x^{2} \cdot - 3$

Multiply $- 3$ by $- 7$ .

$g (x) = - 7 x^{4} + 21 x^{3} + 14 x^{2} x + 14 x^{2} \cdot - 3$

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

Move $x$ .

$g (x) = - 7 x^{4} + 21 x^{3} + 14 (x \cdot x^{2}) + 14 x^{2} \cdot - 3$

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

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

$g (x) = - 7 x^{4} + 21 x^{3} + 14 (x \cdot x^{2}) + 14 x^{2} \cdot - 3$

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

$g (x) = - 7 x^{4} + 21 x^{3} + 14 x^{1 + 2} + 14 x^{2} \cdot - 3$

Add $1$ and $2$ .

$g (x) = - 7 x^{4} + 21 x^{3} + 14 x^{3} + 14 x^{2} \cdot - 3$

Multiply $- 3$ by $14$ .

$g (x) = - 7 x^{4} + 21 x^{3} + 14 x^{3} - 42 x^{2}$

Add $21 x^{3}$ and $14 x^{3}$ .

$g (x) = - 7 x^{4} + 35 x^{3} - 42 x^{2}$

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.

$- 7 x^{4} \to 4$

$35 x^{3} \to 3$

$- 42 x^{2} \to 2$

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.

$- 7 x^{4}$

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.

$- 7 x^{4}$

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

$- 7$

Step 5

List the results.

Polynomial Degree: $4$

Leading Term: $- 7 x^{4}$

Leading Coefficient: $- 7$

Find the Degree, Leading Term, and Leading Coefficient g(x)=-7x^2(x-2)(x-3)

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