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Ch 9 Polynomials 9.1 Polynomials Monomial • A number, variable, or a product of numbers and variables that have only positive exponents • Cannot have an exponent that is a variable Example • Determine whether each expression is a monomial. Explain why or why not. – a2 b3 c 1 – x. – 5z-3 – x2 Polynomial • A monomial or the sum of one or more monomials – Each monomial is a term – Sum: subtraction is just adding the opposite x 3 x 2 3x 2 Special Polynomials • Binomial – – A polynomial with two terms • Trinomial – – A polynomial with three terms Example • State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. – -4x + 2 – 5 + 3x2 + x + 2 – 3x-2 + 4x3 – 5a – 9 + 3 Polynomials • Polynomial terms are arranged in either ascending or descending order – We will arrange our in descending order!! Degree of a Polynomial • Degree of a monomial – – Sum of the exponents of the variables • Degree of a polynomial – – Greatest of the degrees of its terms Example • Find the degree of each polynomial. – 5a2 + 3 – 6x2 – 4x2y – 3xy – 8b4 + 92 – 2ab + 3a2b + 5a4b2 Example • The expression 14x3 – 17x2 – 16x + 34 can be used to estimate the number of eggs that a certain type of female moth can produce. In the expression, x represents the width of the abdomen in millimeters. About how many eggs would you expect this type of moth to produce if her abdomen measures 3 millimeters? Example • The same female moth… how many eggs would you expect her to lay if her abdomen measures 2 millimeters? Assignment • 1st Assignment: – P385: 1 – 3, 5 -15 • 2nd Assignment: – P385: 16 – 69 9.2 Adding and Subtracting Polynomials Adding and Subtracting Polynomials • To add or subtract polynomials, combine all like terms – Remember: like terms have the exact same variables • Including exponents! Example • Find each sum. – (3s + 4t) + (6s – 2t) – (b2 + 4b – 6) + (3b2 – 3b + 1) – (2d2 + 7de – 8e2) + (-d2 + 8e2) – (3x + 9) + (5x + 3) – (7m2 – 6) + (5m – 2) Subtracting Integers • Subtracting is just adding its additive inverse or opposite Example • Find each difference. – (6x + 5) – (3x + 1) – (2g + 7) – (g + 2) – (4a2 – 3a + 4) – (a2 + 6a + 1) – (2y2 – 3y + 5) – (y2 + 2y +8) – (4p2 – p) – (8 + 3p – p2) Example • The measure of the perimeter of a triangle is 9a + 2b. Two of the sides have lengths of 3a + b and 5a. Find the measure of the third side of the triangle. Example • The perimeter of triangle ABC is 7x + 2y. Find the measure of the third side of the triangle. Assignment • 1st Assignment: – P392: 3 - 17 • 2nd Assignment: – P392: 18 – 42, 45 – 51 – 9.1/9.2 Wkst 9.3 Multiplying a Polynomial by a Monomial Multiplying a Polynomial by a Monomial Example • Find each product. – x(x + 1) – g(3g2 + 4) – y(y + 5) – b(2b2 + 3) – -y(2y – 6) – b2(2b2 – 4b – 9) Example • Solve each equation. – 11(y – 3) + 5 = 2(y + 22) – w(w + 12) = w(w + 14) + 12 Example • Solve each equation. – 3(d – 4) – 8 = 5(5d + 1) – 3 – a(3 + a) – 2 = a(a – 1) + 6 Example • Find the area of the shaded region in simplest form. Example • Find the area of the shaded region in simplest form. Assignment • 1st Assignment: – P396: 1, 3 – 16 • 2nd Assignment: – P396: 18 – 61 9.4 Multiplying Binomials Multiplying Binomials • Use distributive property twice!! • (x + 3)(x – 4) Example • Find each product. – (x – 1)(x + 5) – (a + 2)(2a – 3) – (2y – 1)(y – 3) FOIL Example • Find each product – (d + 2)(d + 8) o(5x + y)(4x – 2y) o(n + 3)(n + 5) – (e + 4)(2e – 4) Example • Find the product. – (a – 2)(a2 – 4) – (x2 – 4)(x + 3) Example • The volume V of a rectangular prism is equal to the area of the base B times the height h. Express the volume of the prism as a polynomial. Use V = Bh. Example • Find the volume of a rectangular prism with base dimensions x and x + 4 units and height x + 2 units. Assignment • 1st Assignment: – P402: 4 – 20 • 2nd Assignment: – P402: 22-50 even, 51-65 9.5 Special Products Square of a Sum and Square of a Difference Example • (b + 5)2 • (c – 3)2 • (2d + 1)2 • (3e – 3)2 Real-Life Use • Biologist use a method that is similar to squaring a sum to find the characteristics of offspring based on genetic information. Example • In a certain population, a parent has a 10% chance of passing the gene for brown eyes to its offspring. If an offspring receives one eyecolor gene from its mother and one from its father, what is the probability that an offspring will receive at least one gene for brown eyes? Example • In a certain population, a parent has a 20% chance of passing on a certain gene to its offspring. If an offspring receives one gene each from its mother and father, what is the probability that an offspring will receive at least one of these genes? Product of a Sum and a Difference Example • Find each product. – (3 + a)(3 – a) – (5b – 2)(5b + 2) – (x + y)(x – y) – (5m – 6n)(5m + 6n) Assignment • 1st Assignment: – P408: 1 – 9 • 2nd Assignment: – P408: 10 – 33, 35, 37 – 43 Review • P412: 1 – 59 • P415: 4 – 24