Adding and Subtracting Polynomials Section 0.3

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)

Polynomials Leading Coefficient : Degree: Constant:

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

Classifying Polynomials (a)2 t 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.

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

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

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

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)

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

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

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.

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

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

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

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 + – y – 10y 3 = 4y4y 5 24y 3 – 8y8y y– 20 Combine like terms.

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

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

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

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

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

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