1.4 Polynomials

Polynomials are mathematical expressions with variables and exponents. They're the building blocks of algebra, used to model real-world situations and solve complex problems. Understanding polynomials is crucial for grasping more advanced mathematical concepts.

In this section, we'll cover the basics of polynomials and how to perform operations with them. We'll learn about degree, leading coefficients, and different forms of polynomials. We'll also practice adding, subtracting, and multiplying polynomials, including those with multiple variables.

Polynomial Basics

Degree and leading coefficient

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• Degree of a polynomial is the highest exponent of the variable in the polynomial
  • $3x^4 + 2x^2 - 5x + 1$ has a degree of 4 since the highest exponent is 4
• Leading coefficient is the coefficient of the term with the highest degree
  • $3x^4 + 2x^2 - 5x + 1$ has a leading coefficient of 3, the coefficient of the $x^4$ term

Polynomial Forms and Characteristics

• Standard form of a polynomial arranges terms in descending order of degree (monomial)
• End behavior describes how a polynomial function behaves as x approaches positive or negative infinity
• Polynomials can be divided using polynomial long division, similar to regular long division

Polynomial Operations

Addition and subtraction of polynomials

• Add or subtract polynomials by combining like terms (terms with the same variables and exponents)
  • $(2x^2 + 3x - 1) + (4x^2 - 2x + 5) = 6x^2 + x + 4$ combines like terms $2x^2$ and $4x^2$, $3x$ and $-2x$, and $-1$ and $5$
• Subtract polynomials by distributing the negative sign to each term in the second polynomial before combining like terms
  • $(2x^2 + 3x - 1) - (4x^2 - 2x + 5) = -2x^2 + 5x - 6$ distributes the negative sign to $4x^2$, $-2x$, and $5$, then combines like terms

Multiplication of polynomials

• Multiply polynomials using the distributive property, multiplying each term in the first polynomial by each term in the second polynomial
  • $(2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$ multiplies $2x$ by $x$ and $-4$, and $3$ by $x$ and $-4$, then combines like terms
• FOIL method is a mnemonic for multiplying two binomials: First, Outer, Inner, Last
  • Multiply the first terms, the outer terms, the inner terms, and the last terms, then combine like terms
  • $(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$ multiplies $x$ by $x$, $x$ by $3$, $2$ by $x$, and $2$ by $3$, then combines like terms

Operations with multiple variables

• Apply the same rules for addition, subtraction, and multiplication when working with polynomials containing multiple variables
  • $(2x^2y + 3xy - y) + (4x^2y - 2xy + 5y) = 6x^2y + xy + 4y$ combines like terms $2x^2y$ and $4x^2y$, $3xy$ and $-2xy$, and $-y$ and $5y$
• Use the product rule for exponents when multiplying terms with the same variable in polynomials with multiple variables
  • $(2xy)(3x^2y) = 6x^3y^2$ multiplies the coefficients and adds the exponents of like variables

Simplification of complex polynomials

• Simplify complex polynomial expressions by breaking them down into smaller parts and applying the appropriate operations (addition, subtraction, multiplication)
  • $(3x + 2)(2x - 1) - (4x - 3)(x + 2)$ can be simplified using these steps:
    1. $(3x + 2)(2x - 1) = 6x^2 - 3x + 4x - 2 = 6x^2 + x - 2$ multiplies the binomials
    2. $(4x - 3)(x + 2) = 4x^2 + 8x - 3x - 6 = 4x^2 + 5x - 6$ multiplies the binomials
    3. $(6x^2 + x - 2) - (4x^2 + 5x - 6) = 2x^2 - 4x + 4$ subtracts the resulting polynomials by distributing the negative sign and combining like terms