Find the Degree, Leading Term, and Leading Coefficient p(x)=1/2x^2+3x-4x^3+6x^4-1

$p (x) = \frac{1}{2} x^{2} + 3 x - 4 x^{3} + 6 x^{4} - 1$

Step 1

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

Combine $\frac{1}{2}$ and $x^{2}$ .

$p (x) = \frac{x^{2}}{2} + 3 x - 4 x^{3} + 6 x^{4} - 1$

Simplify the expression.

Move $3 x$ .

$p (x) = \frac{x^{2}}{2} - 4 x^{3} + 6 x^{4} + 3 x - 1$

Move $\frac{x^{2}}{2}$ .

$p (x) = - 4 x^{3} + 6 x^{4} + \frac{x^{2}}{2} + 3 x - 1$

Reorder $- 4 x^{3}$ and $6 x^{4}$ .

$p (x) = 6 x^{4} - 4 x^{3} + \frac{x^{2}}{2} + 3 x - 1$

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.

$6 x^{4} \to 4$

$- 4 x^{3} \to 3$

$\frac{x^{2}}{2} \to 2$

$3 x \to 1$

$- 1 \to 0$

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.

$6 x^{4}$

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.

$6 x^{4}$

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

$6$

Step 5

List the results.

Polynomial Degree: $4$

Leading Term: $6 x^{4}$

Leading Coefficient: $6$