Question

answer the questions about the following polynomial.

\\\frac{1}{2} + 5x^4 - 7x + x^2\\

the expression represents a quadratic polynomial with 4 terms. the constant term is \frac{1}{2}, the leading term is 5x^4, and the leading coefficient is 5.

Explanation:

Identify the polynomial and its degree

The given polynomial expression is:

$$ \frac{1}{2} + 5x^4 - 7x + x^2 $$

To analyze it, we first write it in standard form by arranging the terms in descending order of their exponents:

$$ 5x^4 + x^2 - 7x + \frac{1}{2} $$

The highest exponent of the variable \(x\) is \(4\). Therefore, this is a quartic (or fourth-degree) polynomial.

Count the number of terms

We count the individual terms separated by addition or subtraction:

1. \(5x^4\)
2. \(x^2\)
3. \(-7x\)
4. \(\frac{1}{2}\)

There are \(4\) terms in total.

Determine the constant term

The constant term is the term that does not contain any variable \(x\):

$$ \text{Constant term} = \frac{1}{2} $$

Determine the leading term and leading coefficient

The leading term is the term with the highest degree:

$$ \text{Leading term} = 5x^4 $$

The leading coefficient is the numerical coefficient of this leading term:

$$ \text{Leading coefficient} = 5 $$

Answer:

Answer the questions about the following polynomial.

$$ \frac{1}{2} + 5x^4 - 7x + x^2 $$

The expression represents a <blank>quartic</blank> polynomial with <blank>4</blank> terms. The constant term is <blank>\(\frac{1}{2}\)</blank>, the leading term is <blank>\(5x^4\)</blank>, and the leading coefficient is <blank>5</blank>.