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Published byGwendolyn Allison Modified over 9 years ago
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Polynomials By C. D.Toliver
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Polynomials An algebraic expression with one or more terms –Monomials have one term, 3x –Binomials have two terms, 3x + 4 –Trinomials have three terms, x 2 + 3x + 4
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Review: Collecting Like Terms You simplify polynomial expressions by collecting like terms Like terms have the same variable and the same exponent. 2a, 5a, -7a are like terms 2a, 5b, 6c are not like terms a 3, a 2, a are not like terms Constants are also like terms 2, 0.5, ¼ are like terms
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Review Collecting Like Terms Example 1. Simplify 3x 2 + 5x – 2x 2 + 3x + 7 1x 2 + 8x + 7
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Review: Collecting Like Terms Example 2. Simplify (5a + 7b + 6) + (- 8a - 9b + 5) 5a + 7b + 6 + - 8a + - 9b + + 5 5a + 7b + 6 - 8a - 9b + 5 -3a – 2b + 11
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Review: Collecting Like Terms Example 4. Simplify (5a + 7b + 6) – (8a - 6b + 5) 5a + 7b + 6 - 8a - -6b -+5 5a + 7b + 6 - 8a + 6b - 5 -3a + 13b + 1
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Review: Distributive Property We also learned that parenthesis mean to multiply. We use the distributive property to multiply polynomials The distributive property says: a(b+c) = a(b) + a(c)
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Review: Distributive Property Example 1 Multiply 3(x+y) 3(x) + 3(y) 3x +3y
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Review: Distributive Property Example 2. Multiply 4(2x – 3) 4(2x) + 4(-3) 8x -12
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Review: Distribute and Collect For more complex expressions you may need to distribute and collect like terms. Distribute first Then collect
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Review: Distribute and Collect Example 1. Distribute and Collect 6(x + 5) - 2(2x – 8) 6(x) +6(5) -2(2x) -2(-8)Distribute 6x + 30 – 4x + 16Collect 2x + 46
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Review: Distribute and Collect Example 2. Distribute and Collect 3(2x - 4) + 7(x – 2) 3(2x) + 3(-4) +7(x) + 7(-2)Distribute 6x - 12 + 7x - 14Collect 13x - 26
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Review: Distribute and Collect Example 3. Distribute and Collect 5(y - 3) + 4(6 - 2y) 5(y) +5(-3) +4(6) +4(-2y)Distribute 5y - 15 +24 – 8yCollect -3y + 9
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Multiply Polynomials In the previous examples, we were multiplying polynomials by a monomial, e.g., 3 (x+2) 3 is a monomial x+2 is a polynomial What happens when you multiply two polynomials, e.g., (x + 4)(x+2)?
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Multiply Polynomials We will look at three different methods to multiply polynomials You may prefer one method over another Today we will practice all three methods
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Multiply Polynomials Distributive Method Example 1. Multiply (x + 4)(x + 2) x(x+2) + 4(x+2) Distribute x(x) + x(2) + 4(x) +4(2)Distribute x 2 + 2x + 4x + 8Collect x 2 + 6x + 8
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Multiply Polynomials Vertical Method Example 1. Multiply (x + 4)(x + 2) Rewrite vertically X + 4 X + 2 2x + 8 Multiply x 2 + 4x Multiply x 2 + 6x + 8 Combine
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Multiply Polynomials Box Method Example 1. Multiply (x + 4)(x + 2)= x 2 + 6x + 8 x2x2 2x 4x8 x +2 x +4
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Multiply Polynomials Distributive Method Example 2. Multiply (x - 3)(x + 5) x(x+5) - 3(x+5) Distribute x(x) + x(5) - 3(x) -3(5)Distribute x 2 + 5x - 3x - 15Collect x 2 + 2x - 15
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Multiply Polynomials Vertical Method Example 2. Multiply (x - 3)(x + 5) Rewrite vertically X - 3 X + 5 5x - 15 Multiply x 2 - 3x Multiply x 2 + 2x - 15 Combine
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Multiply Polynomials Box Method Example 2. Multiply (x - 3)(x + 5)= x 2 + 2x - 15 x2x2 -3x 5x-15 x -3 x +5
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Multiply Polynomials Distributive Method Example 3. Multiply (2x + 1)(x - 4) 2x(x-4) + 1(x-4) Distribute 2x(x)+2x(-4)+1(x)+1(-4)Distribute 2x 2 - 8x + 1x -4Collect 2x 2 - 7x - 4
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Multiply Polynomials Vertical Method Example 3. Multiply (2x + 1)(x - 4) Rewrite vertically 2x + 1 x - 4 -8x - 4 Multiply 2x 2 + 1x Multiply 2x 2 - 7x - 4 Combine
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Multiply Polynomials Box Method Example 3. Multiply (2x + 1)(x - 4)= 2x 2 - 7x - 4 2x 2 1x -8x-4 2x +1 x -4
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Multiply Polynomials Distributive Method Example 4. Multiply (2x + 3)(3x - 4) 2x(3x-4)+3(3x-4) Distribute 2x(3x)+2x(-4)+3(3x)+3(-4)Distribute 6x 2 - 8x + 9x - 12Collect 6x 2 + 1x - 12
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Multiply Polynomials Vertical Method Example 4. Multiply (2x + 3)(3x - 4) Rewrite vertically 2X + 3 3X - 4 -8x - 12 Multiply 6x 2 + 9x Multiply 6x 2 + 1x - 12 Combine
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Multiply Polynomials Box Method Example 4. Multiply (2x + 3)(3x - 4)= 6x 2 + 1x - 12 6x 2 9x -8x-12 2x +3 3x -4
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