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Adding and Subtracting Polynomials Section 0.3

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Polynomial A polynomial in x is an algebraic expression of the form: The degree of the polynomial is n (largest exponent) The leading coefficient is ( the coefficient on term with highest exponent) The constant term is (the term without a variable) The polynomial should be written in standard form. (Decreasing order according to exponents)

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Polynomials Leading Coefficient : Degree: Constant: 4 3 -9

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Polynomials Naming a polynomial: 1 term - monomial 2 terms - binomial 3 terms - trinomial 4 or more - terms polynomial Example 2x + 7 has 2 terms so it is called a binomial

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Classifying Polynomials (a)2 t + 7 4 The polynomial cannot be simplified. The degree is 4. The polynomial is a binomial. The polynomial can be simplified. The degree is 2. The simplified polynomial is a monomial. (b)3 e + 5 e – 9 e 222 = – e 2 Two terms. One term.

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Combine like terms and put the polynomial in standard form. What degree is the polynomial? Name the polynomial by the number of terms. Degree is 5 Trinomial

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Adding Polynomials Adding Polynomials Horizontally Add 2n – 7n – 4 and – 5n – 8n + 10. 4343 ( 2n – 7n – 4 ) + ( – 5n – 8n + 10 ) 4343 – 3n 4 – 15n 3 + 6=

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Find the sum (8y – 7y – y + 3) + (6y + 2y – 4y + 1). 3232 + 4– 5y 14y 32 – 5y Adding Polynomials

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Subtracting Polynomials To subtract two polynomials, change all the signs of the second polynomial and add the result to the first polynomial. (Distribute the negative)

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Subtracting Polynomials Perform the subtraction ( 3x – 5 ) – ( 6x – 4 ). ( 3x – 5 ) – ( 6x – 4 ) Change the signs in the second polynomial. – 3x= – 1 = 3x – 5 – 6x + 4

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Subtracting Multivariable Polynomials Add or subtract as indicated. – ab ( 2a b – 4ab + b ) – ( 5a b – 3ab + 7b ) 2 22 2 = 2a b – 4ab + b – 5a b + 3ab – 7b 2 22 2 = – 3a b 2 – 6b 2

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Multiplying Polynomials (a) 5x ( 6x + 7 ) 24 Distributive property = 5x ( 6x ) 24 +5x ( 7 ) 2 = 30x + 35x 62 Multiply monomials. Use the distributive property to find each product.

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Multiplying Polynomials (b) – 2h ( – 3h + 8h – 1 ) 492 Use the distributive property to find each product.

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Multiplying Binomial times Binomial F ( 3g + 2 ) ( 9g – 4 ) O I L 3g ( 9g )Multiply the First terms: 3g ( – 4 )Multiply the Outer terms: 2 ( 9g )Multiply the I nner terms: 2 ( – 4 )Multiply the Last terms: = 27g – 12g + 18g – 8 2 = 27g + 6g – 8 2 FOIL

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Multiplying Polynomials ( 6a + 3b ) ( 4a – 2b ) = 24a – 6b 2 2

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Multiplying Binomial times Trinomial (Megafoil) Distributive property Multiply ( 2y – 5 )( 2y – 7y + 4 ). 2 3 ( 2y – 5 )( 2y – 7y + 4 ) 2 3 = (2y ) 2 3 (–7y) (2y ) 2 + (4) (2y ) 2 + (–7y) + (–5)(4) + (–5) (2y ) 3 + = 4y4y 5 14y 3 – 8y8y 2 + – 20 + 35y – 10y 3 = 4y4y 5 24y 3 – 8y8y 2 ++ 35y– 20 Combine like terms.

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Square a binomial (x+4)² (x+4)(x+4) x² + 4x + 4x + 16 x² + 8x + 16 (x-7)² (x-7)(x-7) x² - 7x - 7x + 49 x² - 14x + 49

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Square the binomial (2a-3b)² (2a-3b)(2a-3b) 4a² - 6ab - 6ab + 9b² 4a² - 12ab + 9b²

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Find the product (x+7)(x-7) x² - 7x + 7x – 49 x² - 49 (2x - ½)(2x + ½) 4x² + x – x - ¼ 4x² - ¼

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Simplify as much as possible -(2x – 6)² -(2x – 6) (2x – 6) -(4x² - 12x – 12x + 36) -(4x² - 24x + 36) -4x² + 24x – 36 3(2x – 4y)² 3(2x – 4y) (2x – 4y) 3(4x² - 8xy – 8xy + 16y²) 3(4x² - 16xy + 16y²) 12x² - 48xy + 48y²

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Cubing a Binomial (x + 4)³ = (x + 4) (x + 4) (x + 4) = (x + 4)(x² + 8x + 16) = x(x²) + x(8x) + x(16) + 4(x²) + 4(8x) + 4(16) = x ³ + 8x² + 16x + 4x² + 32x + 64 = x ³ + 12x² + 48x + 64

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Cubing a Binomial (2x – 3)³ (2x – 3)(2x – 3)(2x – 3) (2x – 3)(4x² - 12x + 9) 2x(4x²) + 2x(-12x) + 2x(9) – 3(4x²) – 3(-12x) – 3(9) 8x³ - 24x² + 18x – 12x² + 36x – 27 8x³ - 36x² + 54x – 27

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