2.2 - Polynomials in One Variable

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Introduction to Polynomials

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Teacher
Teacher

Today we will learn about polynomials in one variable. Can anyone tell me what a polynomial is?

Student 1
Student 1

Isn't it just a mathematical expression involving sums and products?

Teacher
Teacher

Yes, but specifically, a polynomial is an expression that includes terms like these: ax^n, where a is a coefficient and n is a non-negative integer. Can anyone give an example?

Student 2
Student 2

x² + 5x + 6 is a polynomial!

Teacher
Teacher

Exactly! And how many terms does this polynomial have?

Student 3
Student 3

It has three terms!

Teacher
Teacher

Excellent! So we classify polynomials by the number of terms. We've got monomials, binomials, and trinomials. Remember this with the acronym M-B-T!

Student 4
Student 4

Got it! M stands for monomials, B for binomials, and T for trinomials!

Teacher
Teacher

Great! Let's move on to the degrees of these polynomials.

Recap and Closing

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Teacher
Teacher

Let's summarize what we've learned today about polynomials in one variable. What are the types of polynomials?

Student 1
Student 1

Monomials, binomials, and trinomials!

Student 2
Student 2

And we discussed the degrees and zeroes of polynomials too.

Teacher
Teacher

That's right! Remember that the degree tells us about the polynomial's leading behavior and that zeros are vital for factorization. Excellent class today, everyone!

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

The section provides an overview of polynomials in one variable, introducing key definitions, types, and properties, including degrees and coefficients.

Standard

This section discusses polynomials in one variable, detailing their components such as terms, coefficients, and degrees. It classifies polynomials based on the number of terms and their degrees, further explaining the significance of zeroes in polynomial equations. The section concludes by emphasizing the importance of polynomials in algebraic expressions and their applications.

Detailed

Detailed Summary

In this section, we explore polynomials in one variable, defined as algebraic expressions of the form:

Polynomials

Where:
- a₀, a₁, a₂, ..., aₙ are constants (coefficients)
- x is the variable
- n is a non-negative integer (degree of the polynomial)

The expression is considered a polynomial if every term is of the form c * (variable) with whole number exponents. We categorize polynomials based on the number of terms:
- Monomial: A polynomial with one term (e.g., 5x).
- Binomial: A polynomial with two terms (e.g., x + 2).
- Trinomial: A polynomial with three terms (e.g., x² + 3x + 2).

The degree of a polynomial is the highest power of the variable present. For example, in the polynomial 4x^3 + 7x^2 - x + 1, the degree is 3. The section further highlights zeroes of polynomials, explaining that a zero of p(x) is a real number c such that p(c) = 0. The importance of the Factor Theorem and its application in identifying factors of polynomials based on their zeroes is also discussed, along with the use of polynomial equations in algebraic problem solving.

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Understanding Variables and Constants

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Let us begin by recalling that a variable is denoted by a symbol that can take any real value. We use the letters x, y, z, etc. to denote variables. Notice that 2 x, 3x, – x, –1/2x are algebraic expressions. All these expressions are of the form (a constant) × x.

Detailed Explanation

In mathematics, a variable is a symbol that represents a number that can vary or change. For example, in the expression '3x', 'x' is a variable, while '3' is a constant multiplier. Variables are essential for forming algebraic expressions, which can describe situations in mathematics and applied fields. Examples include expressions like '2x' or '–1/2x', which all include a constant and a variable.

Examples & Analogies

Think of a variable as the temperature in a room, which can change depending on whether a heater is on or off. The temperature is not fixed and can take various values, just like 'x' in mathematical expressions.

Forming Algebraic Expressions

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Now suppose we want to write an expression which is (a constant) × (a variable) and we do not know what the constant is. In such cases, we write the constant as a, b, c, etc. So the expression will be ax, say.

Detailed Explanation

When you don't know the exact value of a constant in an expression, you can use letters like 'a', 'b', or 'c' to represent the unknown constants. For example, 'ax' means 'a times x'. This notation helps simplify expressions and allows for more abstract mathematical reasoning.

Examples & Analogies

Imagine a recipe that requires an unknown amount of sugar. You could denote this unknown amount as 's'. So, instead of giving a specific number, you would write the amount of sugar in your recipe as 'sx', where 'x' represents the quantity of whatever you're making.

Examples of Polynomials

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Note that, all the algebraic expressions we have considered so far have only whole numbers as the exponents of the variable. Expressions of this form are called polynomials in one variable. In the examples above, the variable is x. For instance, x3 – x2 + 4x + 7 is a polynomial in x.

Detailed Explanation

Polynomials are algebraic expressions that consist of terms made up of variables raised to whole number exponents, combined with constants. An example of a polynomial is 'x^3 - x^2 + 4x + 7', which indicates that we’re dealing with the variable 'x' raised to various powers (3, 2, and 1, respectively) and it includes the constant term '7'.

Examples & Analogies

Think of a polynomial like a small business’s profit formula where different products contribute different amounts to the total profit, expressed in terms of the quantity sold (the variable). For example, x^3 could represent the profit from selling product x in cubic units, while the other terms represent profits from different aspects of the business.

Terms and Coefficients

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In the polynomial x2 + 2x, the expressions x2 and 2x are called the terms of the polynomial. Similarly, the polynomial 3y2 + 5y + 7 has three terms, namely, 3y2, 5y and 7.

Detailed Explanation

A polynomial consists of several terms, each of which is made up of a coefficient (the number in front) and a variable raised to a power. In the expression '3y^2 + 5y + 7', the terms are '3y^2' (with coefficient 3), '5y' (with coefficient 5), and '7' (which can be seen as 7y^0). Understanding the structure of these terms helps you manipulate polynomials effectively.

Examples & Analogies

Consider a shopping list where each item (term) has a price (coefficient) attached. For example, if you have '3 apples for $2 each', '3 apples' is akin to a term, where '3' is the number of items and '2' is the cost associated with them.

Types of Polynomials

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A polynomial can have any (finite) number of terms. For instance, x150 + x149 + ... + x2 + x + 1 is a polynomial with 151 terms.

Detailed Explanation

Polynomials can vary in complexity. They can have just one term (known as a monomial), two terms (binomial), or three terms (trinomial). However, they can also have many terms, like the example provided where the polynomial consists of 151 terms. The number of terms does not limit the definition of a polynomial.

Examples & Analogies

Think of a student's report card where each subject represents a different term, and the grades in these subjects represent the coefficients. A student could have many subjects; each will contribute to their overall grade (polynomial).

Understanding the Degree of Polynomials

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Now look at the polynomial p(x) = 3x7 - 4x6 + x + 9. What is the term with the highest power of x? It is 3x7. The exponent of x in this term is 7.

Detailed Explanation

The degree of a polynomial is determined by the term with the highest exponent of the variable. In the polynomial '3x^7 - 4x^6 + x + 9', the term '3x^7' has the highest power (7), making the degree of this polynomial equal to 7. Understanding the degree helps in identifying polynomial behavior and rules for polynomial operations.

Examples & Analogies

Imagine a race where cars have different engine powers (exponents). The car with the most powerful engine corresponds to the term with the highest exponent in a polynomial. The more powerful the engine, the faster the car goes, similar to how higher-degree polynomials often exhibit more complex behavior.

Classifying Polynomials by Degree

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A polynomial of degree one is called a linear polynomial. A polynomial of degree two is called a quadratic polynomial. A polynomial of degree three is called a cubic polynomial.

Detailed Explanation

Polynomials are classified based on their degree. Linear polynomials (degree 1) include expressions like '2x + 3', quadratic polynomials (degree 2) include expressions like 'x^2 - 4x + 4', and cubic polynomials (degree 3) include expressions like 'x^3 - 3x^2 + x - 5'. This classification aids in understanding their characteristics and plotting them on graphs.

Examples & Analogies

Just like a carpenter classifies wood into different grades based on its quality (like first-grade, second-grade, etc.), polynomials are categorized based on their highest powers, helping to determine how they behave mathematically, similar to how different grades of wood will behave differently when used in construction.