Find the Degree, Leading Term, and Leading Coefficient f(x)=(x+1)(x-2)(x-4)

$f (x) = (x + 1) (x - 2) (x - 4)$

Step 1

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

Expand $(x + 1) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$f (x) = (x (x - 2) + 1 (x - 2)) (x - 4)$

Apply the distributive property.

$f (x) = (x \cdot x + x \cdot - 2 + 1 (x - 2)) (x - 4)$

Apply the distributive property.

$f (x) = (x \cdot x + x \cdot - 2 + 1 x + 1 \cdot - 2) (x - 4)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = (x^{2} + x \cdot - 2 + 1 x + 1 \cdot - 2) (x - 4)$

Move $- 2$ to the left of $x$ .

$f (x) = (x^{2} - 2 \cdot x + 1 x + 1 \cdot - 2) (x - 4)$

Multiply $x$ by $1$ .

$f (x) = (x^{2} - 2 x + x + 1 \cdot - 2) (x - 4)$

Multiply $- 2$ by $1$ .

$f (x) = (x^{2} - 2 x + x - 2) (x - 4)$

Add $- 2 x$ and $x$ .

$f (x) = (x^{2} - x - 2) (x - 4)$

Expand $(x^{2} - x - 2) (x - 4)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = x^{2} x + x^{2} \cdot - 4 - x \cdot x - x \cdot - 4 - 2 x - 2 \cdot - 4$

Simplify terms.

Simplify each term.

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

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

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

$f (x) = x^{2} x + x^{2} \cdot - 4 - x \cdot x - x \cdot - 4 - 2 x - 2 \cdot - 4$

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

$f (x) = x^{2 + 1} + x^{2} \cdot - 4 - x \cdot x - x \cdot - 4 - 2 x - 2 \cdot - 4$

Add $2$ and $1$ .

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

Move $- 4$ to the left of $x^{2}$ .

$f (x) = x^{3} - 4 \cdot x^{2} - x \cdot x - x \cdot - 4 - 2 x - 2 \cdot - 4$

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

Move $x$ .

$f (x) = x^{3} - 4 x^{2} - (x \cdot x) - x \cdot - 4 - 2 x - 2 \cdot - 4$

Multiply $x$ by $x$ .

$f (x) = x^{3} - 4 x^{2} - x^{2} - x \cdot - 4 - 2 x - 2 \cdot - 4$

Multiply $- 4$ by $- 1$ .

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

Multiply $- 2$ by $- 4$ .

$f (x) = x^{3} - 4 x^{2} - x^{2} + 4 x - 2 x + 8$

Simplify by adding terms.

Subtract $x^{2}$ from $- 4 x^{2}$ .

$f (x) = x^{3} - 5 x^{2} + 4 x - 2 x + 8$

Subtract $2 x$ from $4 x$ .

$f (x) = x^{3} - 5 x^{2} + 2 x + 8$

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.

$x^{3} \to 3$

$- 5 x^{2} \to 2$

$2 x \to 1$

$8 \to 0$

The largest exponent is the degree of the polynomial.

$3$

Step 3

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

$x^{3}$

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.

$x^{3}$

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

$1$

Step 5

List the results.

Polynomial Degree: $3$

Leading Term: $x^{3}$

Leading Coefficient: $1$

Find the Degree, Leading Term, and Leading Coefficient f(x)=(x+1)(x-2)(x-4)

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