Polynomials Notes PDF
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This document includes notes on polynomials, covering topics such as defining polynomials, different types of polynomials (monomials, binomials, trinomials, and polynomials with more terms), identifying like terms, examples on how to combine like terms, and methods on how to simplify mathematical expressions using the concept of like terms.
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Polynomials https://www.youtube.com/watch?v=jLCftTjDLO4 1 What is a Polynomial? 3x + 2y - 10 What is a variable? A letter which represents an unknown number What is a coefficient? A number multip...
Polynomials https://www.youtube.com/watch?v=jLCftTjDLO4 1 What is a Polynomial? 3x + 2y - 10 What is a variable? A letter which represents an unknown number What is a coefficient? A number multiplying with a variable What is a constant? A number without a variable 2 3x + 2y - 10 What is a term? A number or a product of a number and variable(s) Ex) 3x, 5, a, What are the types of polynomials? To be a polynomial, the variables must only have whole number exponents We can classify polynomials by the number of terms Monomial: 1 term ex) 10, 5x, -0.5xyz Binomial: 2 terms ex) 3x + 2y Trinomial: 3 terms ex) 4x + 2y – 3 Polynomial: more than 3 terms ex) 3𝑥 2 𝑦 + 8𝑥 − 4𝑦 + 10 3 Which of the following are polynomials? 1 1 5𝑥 −2 + 8𝑥 3 𝑥 −8 + 3𝑥 2 𝑥 4𝑥𝑦𝑧 − 5𝑥𝑦 2 2𝑥 𝑥+1 (𝑥 − 2)2 5 𝑥3 + 4 5𝑥 2 + 9𝑥 − 7 4 Degree of a term: The sum of the exponents on the variables Leading Term: The term with the highest degree Degree of a polynomial: Same as the degree of the leading term. 3x + 2x4 – 10 4x3 – 2x2 10x 5 Examples: Arrange the terms in descending powers order of degree. Note: If more than one term has 7 – 3x4 + 4xy2 + 2x3y – 2x2 the same degree, order them alphabetically 5xy – 6 + 10x2y2 + 8x4y + 5x6 6 Examples: Create a polynomial that has: 3 different variables, 3 terms, a degree of 3, a constant term of 5. Arrange the terms in descending order of degree. 9 – 2x5 + 4xy3 + y + 2x3y – 6x4 7 Algebra Tiles X2 X 1 -X2 -X -1 Represent 2x2 - 4x – 3 using the tiles. 8 Example: X2 X 1 -X2 -X -1 Represent 2x2 + 4x – 8x2 + 6x in its simplified form using algebra tiles. 9 What are Like Terms? Terms that have the same variable. 3a -6a -a Simplify means: Write the expression in the simplest form. 10 What to Do: Look at one variable and combine all of the numbers that have that same variable and exponent. Remember to watch the signs in front of the variables. 11 What to Do: Do the same for the rest of the variables Remember to watch the signs in front of the variables. 12 Example: 4x2 + 3xy – 14x + 7xy + x2 13 Example: Simplify x2 + 4x + 3 + 2x2 + 5x + 1 Simplify 3n – 2n2 – 2n – 4n + 3n2 Simplify 2.8x2 - 4x + 0.3 + 1.4x2 + 5.7x 14 Example: 11 2 3 1 Simplify x + 𝑦 − 𝑥 − 𝑦 + 10 4 3 5 3 15 5.2 Adding and Subtracting Polynomials 16 Adding and Subtracting Polynomials: Simplify (4x2 + 3xy – 2y2) + (-x2 + 7y2) 17 Example: Simplify (x2 + 4x + 3) - (2x2 - 5x + 1) 18 Example: Simplify (-2x2 + 7) – (-9x – 5x - 1) 19 Example: (9x2 + 7xy – 4y2 ) + (4x2 + 3xy – 2y2) (x2 - 6xy + 7y2) – (x2 - 2xy + 7y2) 20 Example: Simplify (-2x2 + 7) + [-9x – (5x - 1)] – (x2 + 3) 21 Example: Simplify: 1 2 2 1 3 2 1 7 𝑥 + 𝑥 − − 𝑥 − 𝑥 + 4 3 5 8 9 10 22 Example: Write a simplified expression for the perimeter of the triangle. 23 Example: Simplify an expression for the perimeter of the shape: 24 Example: Find the surface area of the shape. 25 Example: Determine the circumference of all three circles together. Leave your answers in terms of pi. Homework: Workbook p.175 #7odds, 8a-e, 9(challenge!) 26
