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7.0 day 1: Polynomial Arithmetic Adding 2) Subtracting 3) Multiplying Factoring the GCF 1) 4) 1 Adding and Subtracting Polynomials The degree of a polynomial is the greatest of the degrees of any of its terms. The degree of a term is the sum of the exponents of the variables. Examples: 3y2 + 5x + 7 degree 2 21x5y + 3x3 + 2y2 degree 6 Common polynomial functions are named according to their degree. Function linear Equation f (x) = mx + b Degree one quadratic f (x) = ax2 + bx + c, a 0 two cubic f (x) = ax3 + bx2 + cx + d, a 0 three 2 Adding and Subtracting Polynomials To add polynomials, combine like terms. Examples: Add (5x3 + 6x2 + 3) + (3x3 – 12x2 – 10). Use a horizontal format. (5x3 + 6x2 + 3) + (3x3 – 12x2 – 10) = (5x3 + 3x3 ) + (6x2 – 12x2) + (3 – 10) = 8x3 – 6x2 – 7 Rearrange and group like terms. Combine like terms. 3 Adding and Subtracting Polynomials Add (6x3 + 11x –21) + (2x3 + 10 – 3x) + (5x3 + x – 7x2 + 5). Use a vertical format. 6x3 + 11x – 21 2x3 – 3x + 10 5x3 – 7x2 + x + 5 13x3 – 7x2 + 9x – 6 Arrange terms of each polynomial in descending order with like terms in the same column. Add the terms of each column. 4 Adding and Subtracting Polynomials The additive inverse of the polynomial x2 + 3x + 2 is – (x2 + 3x + 2). This is equivalent to the additive inverse of each of the terms. – (x2 + 3x + 2) = – x2 – 3x – 2 To subtract two polynomials, add the additive inverse of the second polynomial to the first. 5 Adding and Subtracting Polynomials Example: Add (4x2 – 5xy + 2y2) – (–x2 + 2xy – y2). (4x2 – 5xy + 2y2) – (– x2 + 2xy – y2) = (4x2 – 5xy + 2y2) + (x2 – 2xy + y2) = (4x2 + x2) + (– 5xy – 2xy) + (2y2 + y2) = 5x2 – 7xy + 3y2 Rewrite the subtraction as the addition of the additive inverse. Rearrange and group like terms. Combine like terms. 6 Multiplying Polynomials To multiply a polynomial by a monomial, use the distributive property and the rule for multiplying exponential expressions. Examples:. Multiply: 2x(3x2 + 2x – 1). = 2x(3x2 ) + 2x(2x) + 2x(–1) = 6x3 + 4x2 – 2x 7 Multiplying Polynomials Multiply: – 3x2y(5x2 – 2xy + 7y2). = – 3x2y(5x2 ) – 3x2y(–2xy) – 3x2y(7y2) = – 15x4y + 6x3y2 – 21x2y3 8 Multiplying Polynomials To multiply two polynomials, apply the distributive property. Don’t forget to apply properties of exponents!!! Example: Multiply: (x – 1)(2x2 + 7x + 3). = (x – 1)(2x2) + (x – 1)(7x) + (x – 1)(3) = 2x3 – 2x2 + 7x2 – 7x + 3x – 3 = 2x3 + 5x2 – 4x – 3 9 Multiplying Polynomials Example: Multiply: (x – 1)(2x2 + 7x + 3). Two polynomials can also be multiplied using a vertical format. Example: 2x2 + 7x + 3 x–1 – 2x2 – 7x – 3 2x3 + 7x2 + 3x 2x3 + 5x2 – 4x – 3 Multiply – 1(2x2 + 7x + 3). Multiply x(2x2 + 7x + 3). Add the terms in each column. 10 Multiplying Polynomials To multiply two binomials use a method called FOIL, which is based on the distributive property. The letters of FOIL stand for First, Outer, Inner, and Last. 1. Multiply the first terms. 4. Multiply the last terms. 2. Multiply the outer terms. 5. Add the products. 3. Multiply the inner terms. 6. Combine like terms. 11 Multiplying Polynomials Examples: Multiply: (2x + 1)(7x – 5). First Outer Inner Last = 2x(7x) + 2x(–5) + (1)(7x) + (1)(–5) = 14x2 – 10x + 7x – 5 = 14x2 – 3x – 5 12 Multiplying Polynomials Multiply: (5x – 3y)(7x + 6y). First Outer Inner Last = 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y) = 35x2 + 30xy – 21yx – 18y2 = 35x2 + 9xy – 18y2 13 Special Cases for Multiplication The multiply the sum and difference of two terms, use this pattern: (a + b)(a – b) = a2 – ab + ab – b2 = a2 – b2 square of the second term square of the first term 14 Special Cases for Multiplication Examples: (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (x + 1)(x – 1) = (x)2 – (1)2 = x2 – 1 15 Special Cases for Multiplication To square a binomial, use this pattern: (a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2 square of the first term twice the product of the two terms square of the last term 16 Special Cases for Multiplication Examples: Multiply: (2x – 2)2 . = (2x)2 + 2(2x)(– 2) + (– 2)2 = 4x2 – 8x + 4 Multiply: (x + 3y)2 . = (x)2 + 2(x)(3y) + (3y)2 = x2 + 6xy + 9y2 17 Example: The length of a rectangle is (x + 5) ft. The width is (x – 6) ft. Find the area of the rectangle in terms of the variable x. x–6 A = L · W = Area L = (x + 5) ft W = (x – 6) ft x+5 A = (x + 5)(x – 6 ) = x2 – 6x + 5x – 30 = x2 – x – 30 The area is (x2 – x – 30) ft2. 18 Find the greatest common factor of a list terms. of Recall that to factor means “to write a quantity as a product.” For example, Multiplying Factoring 6 · 2 = 12 12 = 6 · 2 Factors Product Product Factors other factored forms of 12 are − 6(−2), 3 · 4, −3(−4), 12 · 1, and −12(−1). More than two factors may be used, so another factored form of 12 is 2 · 2 · 3. The positive integer factors of 12 are 1, 2, 3, 4, 6, 12. Find the greatest common factor of a list of terms. An integer that is a factor of two or more integers is called a common factor of those integers. For example, 6 is a common factor of 18 and 24. Other common factors of 18 and 24 are 1, 2, and 3. The greatest common factor (GCF) of a list of integers is the largest common factor of those integers. Thus, 6 is the greatest common factor of 18 and 24. Recall 1 that a prime number has only itself and 1 as factors. Find the greatest common factor of a list of terms. (cont’d) Factors of a number are also divisors of the number. The greatest common factor is actually the same as the greatest common divisor. The are many rules for deciding what numbers to divide into a given number. Here are some especially useful divisibility rules for small numbers. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the greatest common factor of a list of terms. (cont’d) Find the greatest common factor (GCF) of a list of numbers as follows. Step 1: Factor. write each number in a prime factored form. Step 2: List common factors. List each prime number that is a factor of every number in the list. (If a prime does not appear in one of the prime factored forms, it cannot appear in the greatest common factor.) Step 3: Choose least exponents. Use as exponents on the common prime factors the least exponent from the prime factored forms. Step 4: Multiply. Multiply the primes from Step 3. If there are no primes left after Step 3. The greatest common factor is 1. EXAMPLE 1 Finding the Greatest Common Factor for Numbers Find the greatest common factor for each list of numbers. Solution: 50, 75 50 2 5 5 GCF = 25 75 3 5 5 12, 18, 26, 32 12 2 2 3 GCF = 2 12, 13, 14 GCF = 1 18 2 3 3 12 2 2 3 13 113 26 2 13 32 2 2 2 2 2 14 2 7 EXAMPLE 2 Finding the Greatest Common Factor for Variable Terms Find the greatest common factor for each Solution: list of terms. 16r 9 , 10r15 , 8r12 GCF = 2r 9 s 4t 6 , s3t 6 , s9t 2 GCF = s 3t 2 x 2 y 3 , xy 5 GCF = xy 3 or xy 3 16r 9 1 2 2 2 2 r 9 10r15 1 2 5 r15 8r12 2 2 2 r12 s 4t 6 s 4 t 6 s 3t 6 s 3 t 6 s 9t 2 s 9 t 2 x 2 y 3 1 x 2 y 3 xy5 1 x y5 Factor out the greatest common factor. Writing a polynomial (a sum) in factored form as a product is called factoring. For example, the polynomial 3m + 12 has two terms: 3m and 12. The GCF of these terms is 3. We can write 3m + 12 so that each term is a product of 3 as one factor. 3m + 12 = 3 · m + 3 · 4 = 3(m + 4) Distributive Property. The factored form of 3m + 12 is 3(m + 4). This process is called factoring out the greatest common factor. The polynomial 3m + 12 is not in factored form when written as 3 · m + 3 · 4. The terms are factored, but the polynomial is not. The factored form of 3m +12 is the product 3(m + 4). EXAMPLE 3 Factoring Out the Greatest Common Factor Factor out the GCF. In the fifth example, use fractions in theSolution: factored form. 6 x 12 x 4 6x 2 x 2 2 2 5t 4 6t 2 5t 2 30t 6 25t 5 10t 4 r r 12 r10 r 2 1 10 8 p q 16 p q 12 p q 5 2 6 3 1 9 3 2 x x 4 4 4 7 4 p q 2 p 4 p q 3q 4 2 2 5 1 2 7 x x 3 4 Be sure to include the 1 in a problem like r12 + r10. Always check that the factored form can be multiplied out to give the original polynomial. EXAMPLE 4 Factoring Out the Greatest Common Factor Factor out the greatest common factor. 6 p q r p q Solution: p q 6 r y 4 y 3 4 y 3 4 y 3 y 4