A. A in the variable, x, is a single term or a sum of terms...

a. A ______ in the variable, $x$, is a single term or a sum of terms of the form $a x^{n}$, where $a$ is a real number and $n$ is a nonnegative integer. b. Given the term $a x^{n}, a$ is called the ______ and _______ is called the degree of the term. c. Given the term $x,$ the coefficient of the term is _______ and the degree is _____. d. A monomial is a polynomial with exactly ______ term(s). e. A_______ is a polynomial with exactly two terms. f. A ______ is a polynomial with exactly three terms. g. The term in a polynomial with the highest degree is called the _____ term and its coefficient is called the _______ ______. h. The degree of a polynomial is the _____ degree of all of its terms. i. The degree of a nonzero constant such as 7 is ______ j. If a term of a polynomial has more than yariable, then the degree of the term is the sum of the _____ of the variables contained in the term. k. A ______ function is a function defined by a finite sum of terms of the form $a x^{n},$ where $a$ is a real number and $n$ is a whole number.

a. A ______ in the variable, $x$, is a single term or a sum of terms of the form $a x^{n}$, where $a$ is a real number and $n$ is a nonnegative integer.
b. Given the term $a x^{n}, a$ is called the ______ and _______ is called the degree of the term.
c. Given the term $x,$ the coefficient of the term is _______ and the degree is _____.
d. A monomial is a polynomial with exactly ______ term(s).
e. A_______ is a polynomial with exactly two terms.
f. A ______ is a polynomial with exactly three terms.
g. The term in a polynomial with the highest degree is called the _____ term and its coefficient is called the _______ ______.
h. The degree of a polynomial is the _____ degree of all of its terms.
i. The degree of a nonzero constant such as 7 is ______
j. If a term of a polynomial has more than yariable, then the degree of the term is the sum of the _____ of the variables contained in the term.
k. A ______ function is a function defined by a finite sum of terms of the form $a x^{n},$ where $a$ is a real number and $n$ is a whole number.

Key Concepts

Polynomial Function

A polynomial function is a function defined by a polynomial expression. It maps input values to outputs in a continuous and smooth manner, and its properties are heavily influenced by the degrees and coefficients of its terms.

Multi-variable Term

For a term containing more than one variable, the degree is determined by the sum of the exponents of the variables in that term. This approach allows for a generalized understanding of the concept of degree in polynomials involving multiple variables.

Leading Term and Leading Coefficient

The leading term of a polynomial is the term that has the highest degree, and its coefficient is known as the leading coefficient. This term primarily determines the behavior of the polynomial for large absolute values of the variable.

Trinomial

A trinomial is a polynomial consisting of exactly three distinct terms. Trinomials are a common form in algebra, particularly in quadratic expressions, and their structure is frequently analyzed in factoring and solving equations.

Binomial

A binomial is a type of polynomial that contains exactly two distinct terms. Binomials are often encountered in algebra and are used in various operations such as expansion, factorization, and simplification.

Coefficient

A coefficient is the numerical factor in a term of a polynomial that multiplies the variable. It is a real number that scales the corresponding power of the variable, and it plays a critical role in determining the behavior of the polynomial function.

Degree

The degree of a term in a polynomial is the exponent of its variable, and the degree of the entire polynomial is the highest exponent among its terms. This concept gives insight into the polynomial's end behavior and the number of possible roots.

Monomial

A monomial is a polynomial that consists of only one term. It is expressed in the form a*x^n where a is a constant and n is a nonnegative integer. Monomials provide the simplest building blocks for constructing more complex polynomial expressions.

Polynomial

A polynomial is an algebraic expression formed by a finite sum of terms, each of which consists of a constant coefficient multiplied by a variable raised to a nonnegative integer exponent. Polynomials represent a fundamental class of functions in algebra and serve as the building blocks in many areas of mathematics.

Constant

A nonzero constant in a polynomial is considered to have a degree of zero. Even though it does not contain any variables, it is treated as a polynomial term that is independent of any variable's value.

Transcript

00:01 Here we are given fill in the blanks by using the concept key concept.

00:09 Now part a a polynomial in variable x is defined as a finite sum of terms of the form a x to the power n where a is real number and exponent n is a non -negative integer.

00:27 That is a x to the power n.

00:31 For example 6 x to the power 5 in part a polynomial in the variable x is a single term or the sum of terms of the a x to the power n where a is real number and n is the non -negative integer so answer is polynomial now part b given the term a x to the power n is called the here we see that here here is coefficient and the degree is n now part c given the term x the coefficient of the term is real number a is always real number real number and the degree is always non -negative integer non -negative integer means like 0 1 2, 3 and so on.

02:34 Now in part d, a monomial is a polynomial with exactly fill in the blanks terms.

02:48 We know that if a polynomial has exactly one term it is called monomial.

02:55 For example, 2 x to the power 3.

03:02 Here we see that only one term is given.

03:05 This is monomial.

03:08 So the answer for this part is now part e.

03:26 For this part we know that if a polynomial has exactly two terms it is called binomial.

03:35 So part e answer is binomial.

03:47 For example 2x plus 3 here two terms 2x and 3 there are two terms now part f if a polynomial has exactly three terms it is called trinomial.

04:18 For example, 2x square plus 5x plus this is trinomial.

04:31 For this part, answer is trinomial.

04:43 Part g, the term is a polynomial with highest degree is called.

04:52 We know that the degree of a polynomial is greatest degree of all its terms and coefficient of the greatest degree term is leading coefficient.

05:02 So the answer for this part is leading term and its coefficient is called the leading coefficient.

05:42 For example, suppose we have 2x to the power 5.

05:49 Here we see that leading coefficient is 2 and leading term is 2x to the 4 5...

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