Algebra Examples

h (x) = \frac{5}{3} x^{2} - \sqrt{7 x^{4}} + 8 x^{3} - \frac{1}{2} + x

$h (x) = \frac{5}{3} x^{2} - \sqrt{7 x^{4}} + 8 x^{3} - \frac{1}{2} + x$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine

\frac{5}{3}

$\frac{5}{3}$ and

x^{2}

$x^{2}$ .

h (x) = \frac{5 x^{2}}{3} - \sqrt{7 x^{4}} + 8 x^{3} - \frac{1}{2} + x

$h (x) = \frac{5 x^{2}}{3} - \sqrt{7 x^{4}} + 8 x^{3} - \frac{1}{2} + x$

Step 1.1.2

Rewrite

7 x^{4}

$7 x^{4}$ as

{(x^{2})}^{2} \cdot 7

${(x^{2})}^{2} \cdot 7$ .

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Step 1.1.2.1

Rewrite

x^{4}

$x^{4}$ as

{(x^{2})}^{2}

${(x^{2})}^{2}$ .

h (x) = \frac{5 x^{2}}{3} - \sqrt{7 {(x^{2})}^{2}} + 8 x^{3} - \frac{1}{2} + x

$h (x) = \frac{5 x^{2}}{3} - \sqrt{7 {(x^{2})}^{2}} + 8 x^{3} - \frac{1}{2} + x$

Step 1.1.2.2

Reorder

7

$7$ and

{(x^{2})}^{2}

${(x^{2})}^{2}$ .

h (x) = \frac{5 x^{2}}{3} - \sqrt{{(x^{2})}^{2} \cdot 7} + 8 x^{3} - \frac{1}{2} + x

$h (x) = \frac{5 x^{2}}{3} - \sqrt{{(x^{2})}^{2} \cdot 7} + 8 x^{3} - \frac{1}{2} + x$

h (x) = \frac{5 x^{2}}{3} - \sqrt{{(x^{2})}^{2} \cdot 7} + 8 x^{3} - \frac{1}{2} + x

$h (x) = \frac{5 x^{2}}{3} - \sqrt{{(x^{2})}^{2} \cdot 7} + 8 x^{3} - \frac{1}{2} + x$

Step 1.1.3

Pull terms out from under the radical.

$h (x) = \frac{5 x^{2}}{3} - x^{2} \sqrt{7} + 8 x^{3} - \frac{1}{2} + x$

Step 1.2

Simplify the expression.

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Step 1.2.1

Move

$- \frac{1}{2}$ .

$h (x) = \frac{5 x^{2}}{3} - x^{2} \sqrt{7} + 8 x^{3} + x - \frac{1}{2}$

Step 1.2.2

Move

$- x^{2} \sqrt{7}$ .

$h (x) = \frac{5 x^{2}}{3} + 8 x^{3} - x^{2} \sqrt{7} + x - \frac{1}{2}$

Step 1.2.3

Reorder

$\frac{5 x^{2}}{3}$ and

$8 x^{3}$ .

$h (x) = 8 x^{3} + \frac{5 x^{2}}{3} - x^{2} \sqrt{7} + x - \frac{1}{2}$

Step 2

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

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Step 2.1

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

$8 x^{3} \to 3$

$\frac{5 x^{2}}{3} \to 2$

$- x^{2} \sqrt{7} \to 2$

$x \to 1$

$- \frac{1}{2} \to 0$

Step 2.2

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.

$8 x^{3}$

Step 4

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

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Step 4.1

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

$8 x^{3}$

Step 4.2

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

$8$

Step 5

List the results.

Polynomial Degree:

$3$

Leading Term:

$8 x^{3}$

Leading Coefficient:

$8$