Question

question
determine if the expression $p^5 + 9p^3\sqrt{7} + 10p^4$ is a polynomial or not. if it is a polynomial, state the type and degree of the polynomial.

answer attempt 1 out of 2
the given expression a polynomial.
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Explanation:

Step1: Recall Polynomial Definition

A polynomial in one variable \( p \) is an expression of the form \( a_np^n + a_{n - 1}p^{n - 1}+\dots+a_1p + a_0 \), where \( n \) is a non - negative integer and \( a_i \) are constants (coefficients). The exponents of the variable must be non - negative integers, and the coefficients can be any real numbers (including irrational numbers like \( \sqrt{7} \)).

In the given expression \( p^{5}+9p^{3}\sqrt{7}+10p^{4} \), we can rewrite it as \( 10p^{4}+p^{5}+9\sqrt{7}p^{3} \). The exponents of \( p \) are \( 4 \), \( 5 \), and \( 3 \), which are all non - negative integers. The coefficients are \( 10 \), \( 1 \), and \( 9\sqrt{7} \), which are real numbers.

Step2: Determine if it is a Polynomial

Since the expression satisfies the definition of a polynomial (non - negative integer exponents on the variable and real number coefficients), the given expression is a polynomial.

Step3: Determine the Type and Degree (if needed)

• Type: A polynomial with one variable is called a polynomial in one variable. This is a polynomial in one variable \( p \). Also, since it has three terms, it is a trinomial.
• Degree: The degree of a polynomial in one variable is the highest power of the variable. Here, the powers of \( p \) are \( 5 \), \( 3 \), and \( 4 \). The highest power is \( 5 \), so the degree of the polynomial is \( 5 \).

Answer:

The given expression \(\boldsymbol{\text{is}}\) a polynomial. (It is a trinomial (polynomial in one variable) with degree \( 5 \).)