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Understanding Polynomials: Monomials, Binomials, Trinomials, and Their Properties

This guide delves into the basics of polynomials, defining key concepts such as monomials, binomials, and trinomials. A monomial consists of a single term (e.g., 3x), while a binomial has two terms (e.g., 4y^2 + 3y) and a trinomial has three terms (e.g., 4x^2 + 2x - 3). Learn about polynomial degrees, coefficients, constant and missing terms, and the importance of ordering terms in descending or ascending order. The document also covers collecting like terms to simplify polynomial expressions effectively.

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Understanding Polynomials: Monomials, Binomials, Trinomials, and Their Properties

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  1. 4.3 Polynomials

  2. Monomial: 1 term (axn with n is a non-negative integers) Ex: 3x, -3, or 4y2 • Binomial: 2 terms Ex: 3x - 5, or 4y2 + 3y • Trinomial: 3 terms Ex: 4x2 + 2x - 3

  3. Polynomial: is a monomial or sum of monomials Ex: 4x3 + 4x2 - 2x - 3 or 5x + 2 Identify monomial, binomial, trinomial, or none x4 monomial -2x + 4 binomial monomial 3x2 trinomial -2x2 - 2x +1 None (polynomial) 4x3 + 4x2 - 2x - 3 binomial 4 + (1/2)x

  4. Term: each monomial of the polynomial • Degree: exponents • Degree of polynomial: highest exponent • Coefficient: number in front of variables • Constant term: the term without variable • Missing term: the term that has0 as itscoefficient 0

  5. Ex: -3x4 – 4x2 + x – 1 Term: -3x4 , – 4x2 , x, – 1 Degree 4 2 1 0 Coefficient -3 -4 1 -1 Degree of this polynomial is 4 Constant term: is -1 Missing term (s): is x3

  6. Descending order: exponents decrease from left to right • Ascending order: exponents increase from left to right • When working with polynomials, we often use Descending order

  7. Arrange in descending order using power of x • -6x2 – 8x6 + x8 + 3x - 4 = x8– 8x6 - 6x2 + 3x - 4 Missing terms are: x7, x5, x4, x3 • 5y2 + 4y + 2y4 + 9 = 2y4 +5y2 + 4y + 9 Missing terms are: y3

  8. Collecting Like Terms • Like terms: 4x and 3x 5xy and -6xy 2x2 and x2 When add or subtract like-term, add or subtract only the coefficients of the terms, keep the same variables

  9. 1) -6x4 – 8x3 + 3x - 4 + 5x4 + x3 + 2x2 -7x = -6x4 + 5x4 – 8x3 + x3 + 2x2 + 3x -7x -4 = -x4 - 7x3 + 2x2 - 4x -4 • -6x4 – 8x3 + 3x - 4 - 5x4 - x3 - 2x2 +7x = -6x4 - 5x4 – 8x3 - x3 - 2x2 + 3x +7x - 4 = -11x4 - 9x3 - 2x2 +10x -4

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