Question

1. practice: summarizing fill in the blanks to complete the definition. a monomial usually has the form __, where a is and n is a __. 2) practice: using visual cues complete the labels for the diagram. 9.1.1 study: what is a polynomial? copyright © 2025 apex learning inc. use of this material is subject to apex learning’s terms of use. any unauthorized copying, reuse, or redistribution is prohibited. 1/4

Explanation:

Step1: Recall monomial definition

A monomial is a single term, usually in the form \( a x^n \), where \( a \) is a coefficient (a real number, constant, or a product of constants and variables' coefficients) and \( n \) is a non - negative integer (whole number, including zero). The general form can also be considered as \( a x^{n}\) (or just \( a \) when \( n = 0 \), or \( a x \) when \( n=1 \), etc.), where \( a \) is a constant (or a coefficient) and \( n \) is a non - negative integer, and it represents a single term in a polynomial (or can stand alone as a polynomial of degree \( n \)).

Step2: Fill in the blanks

The first blank (for the form) should be \( a x^{n} \) (or other equivalent forms like \( a x^{n}\) where \( x \) is a variable, but the standard form is \( a x^{n}\)). The second blank (for what \( a \) is) can be "a constant (or a coefficient)" and the third blank (for what \( n \) is) can be "a non - negative integer (or a whole number)". But if we consider the most basic form, the form of a monomial is \( a x^{n} \), where \( a \) is a coefficient (a real number, a constant) and \( n \) is a non - negative integer, and a monomial is a single - term polynomial (or an expression with one term). So putting it together, a monomial usually has the form \( \boldsymbol{a x^{n}} \), where \( a \) is \( \boldsymbol{a \text{ constant (or coefficient)}} \) and \( n \) is \( \boldsymbol{a \text{ non - negative integer}} \), and a monomial is a single - term polynomial (the last blank can be "single - term polynomial" or "a polynomial with one term" or just "a term" in a more basic sense).

Answer:

A monomial usually has the form \(\boldsymbol{a x^{n}}\), where \(a\) is \(\boldsymbol{a \text{ constant (or coefficient)}}\) and \(n\) is \(\boldsymbol{a \text{ non - negative integer}}\), and a monomial is \(\boldsymbol{a \text{ single - term polynomial (or a term)}}\). (The specific blanks would be filled as per the exact requirement of the question, but the key parts are the form \(a x^{n}\), \(a\) as a constant/coefficient, \(n\) as a non - negative integer, and the definition part about being a single - term polynomial or a term.)