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A monomial, in one variable, is the product of a constant times a variable raised to a nonnegative integer power. Thus, a monomial is of the form: ax k where a is a coefficient, x is a variable, and k > 0 is an integer. Monomial 3x 4 2x 9 Coefficient 3 Degree 4 2 1 -9 0 A Binomial in one or more variable is the combination of two monomials separated by addition or subtraction and raised to nonnegative integer powers. Thus, a binomial is of the form ax bx k n where a is a constant, x is a variable, and k > 0 is an integer. Binomial Coefficients 3x 9 x 4 2 x 8x 3 5z 2 Degree 3, 9 4 2 ,8 2 -9, -5 1 A Trinomial in one or more variable is the combination of three monomials separated by addition or subtraction and raised to nonnegative integer powers. Thus, a binomial is of the form ax bx cx k n m where a is a constant, x is a variable, and k > 0 is an integer. Trinomial 3x 9 x 5 x 4 Coefficients Degree 2 2 x 8x 1 2 3 5z 6 z 3 3, 9, 5 2 ,8,1 3, -5, 6 4 2 3 A polynomial in one variable is an algebraic expression of the form: an x n an 1 x n 1 a1 x a0 Where an , an 1..., a1 , a0 are constants, called Coefficients of the Polynomial, n 0 is an integer,and x is a variable. If an 0, it is called the leading Coefficient, and is called the Degree of the Polynomial. n Example: Polynomial: Degree: 2 x 3x x 5 4 2 4 Coefficients: 2, -3, 1, -5 Leading Coefficient: 2 x 2 x x 3x 2 3 Standard Form: (Powers in Descending Order) Degree: (Highest Power) 4 3x4-2x3+x2+x 4 Terms: (Monomials of the Polynomial) x2, x, -2x3, 3z4 Coefficients: (Real Numbers in front of a term) 1, -2, 3 Leading Coefficient: (Coefficient of the Highest Power) 3 Turning Points: (The Highest Power minus one) 3 A Zero Polynomial: (All zero Coefficients) 0x40x3+0x2+0x Addition of Polynomials Combining “like” termsMeaning they must have the same variables all with the same powers/exponents 3 3 2 ( 3 x 2 x 4 ) ( 4 x 3 x 2) Add: You can add by inspection of like terms: (3x 3 2 x 4) (4 x 3 3x 2 2) x 3 3x 2 2 x 2 You can group the “like” terms: (3x 3 4 x 3 ) (3x 2 ) (2 x) (2 4) x 3 3x 2 2 x 2 Line up “like” terms and adding as you would regular numbers: 3x 3 4 x 3 3x 2 2x 4 2 x 3 3x 2 2 x 2 Subtraction of Polynomials Subtract: (5x 2 3x 6) (9 x 2 5x 3) Adding the Opposite- Changing Subtraction to Addition (The same as multiplying it by a Negative one) (5x2-3x+6)+(-9x2+5x+3) You can add by inspection of like terms: (5 x 2 3x 6) (9 x 2 5 x 3) - 4x 2 2x 9 You can now ADD by grouping the “like” terms: (5 x 9 x ) (3x 5 x) (6 3) 2 2 4x 2x 9 2 Line up “like” terms and ADD as you would regular numbers: 5 x 2 3x 6 2 (9 x 5 x 3) 4x 2 2x 9 Multiplication of Polynomials To Multiply a monomial times a polynomial use the Distributive Property of Real Numbers Multiply: 2 x ( x 7 x 10 x 4) 2 3 2 2 x 2 ( x 3 7 x 2 10 x 4) 2 x 2 3 14 x 2 2 20 x1 2 8 x 2 (2 x 5 14 x 4 20 x 3 8x 2 ) To Multiply Two Binomials use the FOIL Method: First, Outer, Inner, Last Multiply: (2 x 3 5) ( x 2 4) (2x3)(x2) + (2x3)(-4) + (5)(x2) + (5)(-4) = 2 x 5 5 x 2 8 x 3 20 To Multiply any two Polynomial s use the Distributive Property of Real Numbers and combine the “like” terms. Multiply: ( x 2 2 x) (4 x 3 2 x 2 3x) (x2)(4x3)+(x2)(-2x2)+(x2)(3x)+(2x)(4x3)+(2x)(-2x2)+(2x)(3x) 4 x 5 2 x 4 3x 3 8 x 4 4 x 3 6 x 2 4 x5 4 x 4 x3 6 x 2 To Multiply any two Polynomial s use the Method For Regular Vertical Multiplication Method with Real Numbers Multiply: ( x 2 2 x) (4 x 3 2 x 2 3x) 4 x 3 2 x 2 3x x2 2x 8x4 - 4x3 + 6x2 4x5 -2x4 +3x3 4x5+ 6x4 – x3 + 6x2 Division of Polynomials When dividing by a Monomial Separate terms over the common denominator and and simplify using Exponent Rules: 10a 5 b 4 2a 3b 2 6a 2 b 2a 2 b Divide: Split into separate terms with the denominator 5 4 3 2 2 10a b 2a b 6a b 2 2 2a b 2a b 2a 2 b 10 5 4 2 1 2 3 2 2 1 6 2 2 1 a b a b a b a b a ba b 2 2 2 5a 5 2 b 4 1 1a 3 2 b 2 1 3a 2 2 b1 1 5a 3b 3 a1b1 3a 0 b 0 5a 3b 3 ab 3 Division of Polynomials When dividing by another Polynomial Find the Quotient and remainder when 2 x 8x x 4 3 2 is divided by x 4 2 x2 1 2 x3 8x2 x 4 2x2 1 using Long Division: x 2 x3 8x 2 x 2 8x2 Check: 2x 2 1 x 4 2 x 8 4 2x 8 2 x 8x x 4 2 x 8 3 2 2 x 8x x 4 3 2 2 x 8x x 4 2x 8 Thus, x4 2 2 2x 1 2x 1 3 2 1234- Synthetic Division: ONLY use when power of divisor is one Write all Coefficients in order of descending Power, if a power is missing write in a zero. Multiply on the diagonals and Add the Columns as you move across the grid. Remember each time you divide by a number the power of the original Polynomial goes down one power. Divide: 3x 4 x x 7 3 2 by x 1 Divisor is x + 1 and the Power is one. x + 1= 0 which gives x = -1 -1 3 4 -3 1 -1 7 0 3 1 0 7 Polynomial now goes down a Power 3x 4 x x 7 x 1 3 2 7 3x x x 1 2