Find the Degree, Leading Term, and Leading Coefficient h(x)=-2x(x-1)(3-x)^2

$h (x) = - 2 x (x - 1) {(3 - x)}^{2}$

Step 1

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

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Simplify the expression.

Multiply $- 1$ by $- 2$ .

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$h (x) = (- 2 x^{2} + 2 x) (9 + 3 (- x) - x \cdot 3 - x (- x))$

Multiply $- 1$ by $3$ .

$h (x) = (- 2 x^{2} + 2 x) (9 - 3 x - x \cdot 3 - x (- x))$

Multiply $3$ by $- 1$ .

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

Rewrite using the commutative property of multiplication.

$h (x) = (- 2 x^{2} + 2 x) (9 - 3 x - 3 x - 1 \cdot (- 1 x \cdot x))$

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

Move $x$ .

$h (x) = (- 2 x^{2} + 2 x) (9 - 3 x - 3 x - 1 \cdot (- 1 (x \cdot x)))$

Multiply $x$ by $x$ .

$h (x) = (- 2 x^{2} + 2 x) (9 - 3 x - 3 x - 1 \cdot (- 1 x^{2}))$

Multiply $- 1$ by $- 1$ .

$h (x) = (- 2 x^{2} + 2 x) (9 - 3 x - 3 x + 1 x^{2})$

Multiply $x^{2}$ by $1$ .

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

Subtract $3 x$ from $- 3 x$ .

$h (x) = (- 2 x^{2} + 2 x) (9 - 6 x + x^{2})$

Expand $(- 2 x^{2} + 2 x) (9 - 6 x + x^{2})$ by multiplying each term in the first expression by each term in the second expression.

$h (x) = - 2 x^{2} \cdot 9 - 2 x^{2} (- 6 x) - 2 x^{2} x^{2} + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

Simplify terms.

Simplify each term.

Multiply $9$ by $- 2$ .

$h (x) = - 18 x^{2} - 2 x^{2} (- 6 x) - 2 x^{2} x^{2} + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

Rewrite using the commutative property of multiplication.

$h (x) = - 18 x^{2} - 2 \cdot (- 6 x^{2} x) - 2 x^{2} x^{2} + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

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

Move $x$ .

$h (x) = - 18 x^{2} - 2 \cdot (- 6 (x \cdot x^{2})) - 2 x^{2} x^{2} + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

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

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

$h (x) = - 18 x^{2} - 2 \cdot (- 6 (x \cdot x^{2})) - 2 x^{2} x^{2} + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

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

$h (x) = - 18 x^{2} - 2 \cdot (- 6 x^{1 + 2}) - 2 x^{2} x^{2} + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

Add $1$ and $2$ .

$h (x) = - 18 x^{2} - 2 \cdot (- 6 x^{3}) - 2 x^{2} x^{2} + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

Multiply $- 2$ by $- 6$ .

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{2} x^{2} + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

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

Move $x^{2}$ .

$h (x) = - 18 x^{2} + 12 x^{3} - 2 (x^{2} x^{2}) + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

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

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{2 + 2} + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

Add $2$ and $2$ .

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{4} + 2 x \cdot 9 + 2 x (- 6 x) + 2 x \cdot x^{2}$

Multiply $9$ by $2$ .

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{4} + 18 x + 2 x (- 6 x) + 2 x \cdot x^{2}$

Rewrite using the commutative property of multiplication.

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{4} + 18 x + 2 \cdot (- 6 x \cdot x) + 2 x \cdot x^{2}$

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

Move $x$ .

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{4} + 18 x + 2 \cdot (- 6 (x \cdot x)) + 2 x \cdot x^{2}$

Multiply $x$ by $x$ .

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{4} + 18 x + 2 \cdot (- 6 x^{2}) + 2 x \cdot x^{2}$

Multiply $2$ by $- 6$ .

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{4} + 18 x - 12 x^{2} + 2 x \cdot x^{2}$

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

Move $x^{2}$ .

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{4} + 18 x - 12 x^{2} + 2 (x^{2} x)$

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

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

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{4} + 18 x - 12 x^{2} + 2 (x^{2} x)$

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

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{4} + 18 x - 12 x^{2} + 2 x^{2 + 1}$

Add $2$ and $1$ .

$h (x) = - 18 x^{2} + 12 x^{3} - 2 x^{4} + 18 x - 12 x^{2} + 2 x^{3}$

Simplify by adding terms.

Subtract $12 x^{2}$ from $- 18 x^{2}$ .

$h (x) = 12 x^{3} - 2 x^{4} + 18 x - 30 x^{2} + 2 x^{3}$

Add $12 x^{3}$ and $2 x^{3}$ .

$h (x) = 14 x^{3} - 2 x^{4} + 18 x - 30 x^{2}$

Simplify the expression.

Move $18 x$ .

$h (x) = 14 x^{3} - 2 x^{4} - 30 x^{2} + 18 x$

Reorder $14 x^{3}$ and $- 2 x^{4}$ .

$h (x) = - 2 x^{4} + 14 x^{3} - 30 x^{2} + 18 x$

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.

$- 2 x^{4} \to 4$

$14 x^{3} \to 3$

$- 30 x^{2} \to 2$

$18 x \to 1$

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.

$- 2 x^{4}$

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.

$- 2 x^{4}$

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

$- 2$

Step 5

List the results.

Polynomial Degree: $4$

Leading Term: $- 2 x^{4}$

Leading Coefficient: $- 2$

Find the Degree, Leading Term, and Leading Coefficient h(x)=-2x(x-1)(3-x)^2

Related Questions

Support LCMGCF.com by disabling your adblocker.