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Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
Polynomial operations (1)
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Polynomial operations (1)

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  • 1. Polynomial Operations Chapter 6 p.333 1
  • 2. 2
  • 3. 3
  • 4. What is polynomial? A polynomial of a letter is an algebraic expression that is the sum of the products between real numbers and the non-negative integer powers of the letter. Examples, Suppose that the letter is x, then 3 x + 2, 2 x 2 − 3 x + 5, x 3 + 2 x 2 − 4 x, x100 are all polynomials of x. Note: 1. The Polynomial letter could be any letter. 3 y + 2, 2t 2 − 3t + 5, r 3 + 2r 2 − 4r , p100 are all polynomials . 2. Any number is also considered as a polynomial. This is because 5 = 5x 0 , which is 5 times the zero power of x. 3. Each product is called the term of the polynomial 2 y 2 − 3 y + 5 has three terms: 2 y 2 , − 3 y, 5. 4
  • 5. 4. Each number of the product is called the coeficient of the polynomial 3 x + 2 has 2 coeficients 3 and 2 2 y 2 − 3 y + 5 has 3 coeficients 2, − 3 and 5. x 3 + 2 x 2 − 4 x − 3, has 4 coeficients 1, 2, − 4 and − 3. x100 has one coeficient 1. 5. The heighest power exponent of x is called the degree of the polynomial. 3 x + 2 has degree =1, 2 y 2 − 3 y + 5 has degree =2 x3 + 2 x 2 − 4 x − 3 x100 has degree =100 5 has degree = 0. 6. A singleton term polynomial is called monomial, such as x100 , 2x, 5 5
  • 6. 7. A two terms polynomial is called binomial, such as 2 x + 5, 9 p 5 + 7, x 2 − 4. 8. A three terms polynomial is called trinomial, such as 2 x 2 − 3 x + 5, 9 p 5 + 7 p 2 + p, x 2 + 4 x + 4. 9. 2 x 2 − 3x + 5 x , 1 + 4 x + 4, 2 x x +1 x−2 are not polynomials. Why? 10. Polynomial could have more than one letters, if all letters have non-negative powers, such as 2 x 2 y − 3 xy 3 + 5 x 6 y 2 , 8m 2 p 5 + 9m3 p 2 They are called multi-varialbles polynomials. In the above examples, we also can consider that only one letter as variable and other letters as numbers. 6
  • 7. Polynomial Operations 1. Add and subtract: We only do on the same power terms. ax n ± bx n = ( a ± b ) x n 3x 3 + 5 x 3 = ( 3 + 5 ) x 3 = 8 x 3 3m 5 − 7 m 5 = ( 3 − 7 ) m 5 = − 4m 5 when we have multiple terms, we just add or subtract terms with the same powers Examples: (2 x 4 − 3 x 2 + 5 x) + ( x 4 + 2 x 2 + 4 x) = (2 + 1) x 4 + (−3 + 2) x 2 + (5 + 4) x = 3x 4 − x 2 + 9 x (3 x 2 + 7 x − 6) − (5 x 2 − 4 x + 8) = (3 − 5) x 2 + (7 − (−4)) x + (−6 − 8) = −2 x 2 + 11x − 14 Can we do addition 2 x 4 + 3 x 2 ? 7
  • 8. Do operation vertically (3 x 2 + 7 x − 6) − (5 x 2 − 4 x + 8) = − 2 x 2 + 11x − 14 3x 2 + 7 x − 6 5 x 2 − 4 x + 8 (− − 2 x 2 + 11x − 14 or (3 x 3 + 7 x 2 − 6) + (5 x 3 − 4 x + 8) = 8 x 3 + 7 x 2 − 4 x + 2 3 x 3 + 7 x 2 + 0 ×x − 6 5 x3 + 0 ×x 2 − 4 x + 8 (+ 8 x3 + 7 x 2 − 4 x + 2 Align the same power terms vetically. Put 0s for the missing power terms. Then do numbers operations veritcally. 8
  • 9. 2. Multiplication: Use formula ( ax ) ×( bx ) m n = abx m + n ( 3x ) ( 5 x ) = ( 3 ×5 ) x = 15x ( 3m ) ( −7m ) = ( 3 ×( −7 ) ) m 2 3+ 2 3 5 2 5 5+ 2 = − 21m7 When two polynomials have multiple terms, then every term of the 1st polynomial must multiply to every term of the 2nd polynoimial. Examples: (3 x − 4)(2 x 2 − 3 x + 5) = (3 x)(2 x 2 − 3 x + 5) + (−4)(2 x 2 − 3 x + 5) = (3 x)(2 x 2 ) + (3 x)(−3 x) + (3 x)(5) + ( −4)(2 x 2 ) + (−4)(−3 x) + (−4)(5) = 6 x 3 − 9 x 2 + 15 x − 8 x 2 + 12 x − 20 = 6 x 3 − 17 x 2 + 27 x − 20 9
  • 10. Do multiplication vertically, put 0 for missing power terms (3 x − 4)(2 x 2 − 3x + 5) = 6 x 3 − 17 x 2 + 27 x − 20 2x2 6 x3 6 x3 −8 x 2 −9 x 2 −17 x 2 −3 x 3x +12 x +15 x +27 x +5 −4 (× −20 (+ −20 (2 x + 3)( x 2 − 5) = 2 x 3 + 3x 2 − 10 x − 15 x2 2 x3 2 x3 3x 2 0 x2 +3 x 2 0 ×x 2x 0x −10 x −10 x −5 +3 (× −15 (+ −15 10
  • 11. Multiplication vertically example2 ( x 2 + 2 x − 3)(2 x 2 − 3x + 5) = 2 x 4 + x 3 − 7 x 2 + 19 x − 15 x2 + 2x −3 2x2 −3 x + 5 (× 5x2 +10 x −15 −3 x 3 −6 x 2 +9 x 2 x4 +4 x 3 −6 x 2 2 x4 + x3 − 7 x2 + 19 x Multiply by 5 Multiply by −3x Multiply by 2x2 − 15 11
  • 12. Vertical multiplication with numbers only (3 x − 4)(2 x 2 − 3 x + 5) = 6 x3 − 17 x 2 + 27 x − 20 (2 x + 3)( x 2 − 5) = 2 x 3 + 3 x 2 − 10 x − 15 12
  • 13. ( x 2 + 2 x − 3)(2 x 2 − 3 x + 5) = 2 x 4 + x 3 − 7 x 2 + 19 x − 15 13
  • 14. Two binomial multiplication. Use FOIL rule First terms, Outside terms, Inside terms, Last terms (2 x − 3)(6 x + 5) = ( 2 x ) ( 6 x ) + ( 2 x ) ( 5 ) + ( −3) ( 6 x ) + ( −3) ( 5 ) 1 24 124 1 24 124 4 3 4 3 4 3 4 3 F O I L = 12 x 2 + 10 x − 18 x − 15 = 12 x 2 − 8 x − 15 or simply vertical way 2x 6x 12 x 2 12 x 2 −3 +5 (× 10 x −15 −18 x − 8x − 15 14
  • 15. Exercises 1. (3 x 2 − 4 x + 5) + ( −2 x 2 + 3 x − 2) 2. (4m3 − 3m 2 + 5) + ( −3m3 − m 2 + 5) 3. 2(12 x − 8 x + 6) − 4(3 x − 4 x + 2) 2 2 4. − (8 x + x − 3) + (2 x + x) − (4 x − 1) 2 2 2 5. 2 x 3 (3 x 2 − 5 x + 2) 6. ( x 2 + 5)(3 x − 2) 7. (4 x + 5)(3 x − 2) 8. (3 x 2 − 4 x + 5)(3 x + 1) 9. (3 x 2 − 4 x + 5)(2 x 2 + x − 2) 15
  • 16. Some important formulas 1. ( x + y )( x − y ) = x 2 − y 2 (sum and difference product) Use FOIL ( x + y )( x − y ) = x 2 − xy + xy − y 2 = x2 − y 2 or (a + b)(a − b) = a 2 − b 2 2. ( x + y ) 2 = x 2 + 2 xy + y 2 (square of sum ) Use FOIL ( x + y ) 2 = ( x + y )( x + y ) = x 2 + xy + xy + y 2 = x 2 + 2 xy + y 2 or (a + b) 2 = a 2 + 2ab + b 2 3. ( x − y ) 2 = x 2 − 2 xy + y 2 (square of difference ) Use FOIL ( x − y ) 2 = ( x − y )( x − y ) = x 2 − xy − xy + y 2 = x 2 − 2 xy + y or (a − b) 2 = a 2 − 2ab + b 2 16
  • 17. 4. ( x + y )( x 2 − xy + y 2 ) =x3 + y 3 (sum of cubic powers) This is because ( x + y )( x 2 − xy + y 2 ) = x( x 2 − xy + y 2 ) + y ( x 2 − xy + y 2 ) ( ) ( ) = x 3 − x 2 y + xy 2 + yx 2 − xy 2 + y 3 = x3 + y 3 Eexamples ( a + 1)( a 2 − a + 1) = a 3 + 1 (a + 2)(a 2 − 2a + 4) = ( a + 2)( a 2 − 2 ×a + 2 2 ) = a 3 + 23 = a 3 + 8 (a + 3)(a 2 − 3a + 9) = (a + 2)(a 2 − 3 ×a + 32 ) = a 3 + 33 = a 3 + 27 5. ( x − y )( x 2 + xy + y 2 ) =x 3 − y 3 (difference of cubic powers) This is because, we can use − y to replace y in the above formula ( x + ( − y ) ) ( x 2 − x ( − y ) + ( − y ) 2 ) =x 3 + ( − y ) 3 which is Examples ( x − y )( x 2 + xy + y 2 ) =x 3 − y 3 (a − 1)(a 2 + a + 1) = a 3 − 1 (a − 2)(a 2 + 2a + 4) = (a − 2)(a 2 + 2 ×a + 2 2 ) = a 3 − 23 = a 3 − 8 (a − 3)(a 2 + 3a + 9) = (a − 2)(a 2 + 3 ×a + 32 ) = a 3 − 33 = a 3 − 27 17
  • 18. Example 1. (3 p + 11)(3 p − 11) = ( 3 p ) − 112 = 9 p 2 − 121 2 2. (5m − 3)(5m + 3) = ( 5m 3 3 ) 3 2 − 32 = 25m6 − 9 3. (9k − 11r )(9k + 11r ) = ( 9k ) − ( 11r 3 2 3 ) 3 2 = 81k 2 − 121r 6 4. (2m + 5) 2 = ( 2m ) + 2( 2m)(5) + ( 5 ) = 4 m 2 + 20 m + 25 2 ( x + y )2 5. ( 3x − 7 y 2 x2 ) = ( 3x ) 4 2 ( a −b ) 2 a2 y2 2 xy 2 − 2(3 x)(7 y )+ ( 7 y 4 2 ab ) 4 2 b2 = 9 x 2 − 42 xy 4 + 49 y 8 6. ( 3 x − 2 y ) ( 9 x + 6 xy + 4 y ) = ( 3 x ) − ( 2 y ) = 27 x 3 − 8 y 6 1 24 144 2444 { 4 3 4 3 1 3 2 2 (a − b) 2 2 a 2 + ab + b 2 3 4 a3 2 3 b3 18
  • 19. Exercises 1. (3 x + 5)(3 x − 5) 2 2 2. (2m3 + n)(2m3 − n) 3. ( 5r + 4t ) 2 2 4. ( 2 x − 3 y ) 4 2 5. (3 p + 5) 2 6. (4 − x)(16 + 4 x + x 2 ) 7. (2a + 3b)(4a 2 − 6ab + 9b 2 ) 19
  • 20. Higher Power of binomial We have (a + b) 2 = a 2 + 2ab + b 2 what is (a + b)3 ? After calculating ( a + b) 3 = (a + b)( a + b) 2 = (a + b)( a 2 + 2 ab + b 2 ) = a (a 2 + 2ab + b 2 ) + b( a 2 + 2ab + b 2 ) = (a 3 + 2a 2b + ab 2 ) + (a 2b + 2a 2b + b3 ) = a 3 + 3a 2b + 3ab 2 + b3 We can see the powers of a is decreasing and the powers of b is increasing. The coefficients are 1, 3, 3, 1. Similarly, (a + b) 4 = (a + b)(a + b)3 = (a + b)(a 3 + 3a 2b + 3a 2b + b 3) = a (a 3 + 3a 2b + 3a 2b + b3) + b(a 3 + 3a 2b + 3a 2b + b 3) = a 4 + 4a 3b + 6a 2b 2 + 4ab3 + b 4 The coefficients are 1, 4, 6, 4, 1. 20
  • 21. Pascal Triangle We see that ( a + b ) has coefficients 1, 1 (a + b) 2 = a 2 + 2ab + b 2 has coefficients 1, 2, 1 ( a + b)3 = a 3 + 3a 2b + 3a 2b + b3 has coefficients 1, 3, 3,1 (a + b) 4 = a 4 + 4a 3b + 6a 2b 2 + 4ab3 + b 4 has coefficients 1, 4, 6, 4, 1 So we can arrage them into a triangle like 1 1 1 1 1 2 3 4 1 3 6 1 4 1 Every number is the sum of two numbers on its shoulder. 21
  • 22. 1 1 1 1 1 1 2 3 4 5 1 1 3 6 10 1 4 10 1 5 1 It we continue to calculate the numbers on the next line, we will get numbers 1, 5, 10, 10, 5, 1, which are coefficients of power (a + b)5 . Therefore we get (a + b)5 = a 5 + 5a 4b + 10a 3b 2 + 10a 2b3 + 6ab 4 +b 5 This triangle is called the Pascal Triangle. 22
  • 23. Powers of (a − b) n Similarly, we have (a − b) 2 = a 2 − 2ab + b 2 (a − b)3 = a 3 − 3a 2b + 3a 2b − b3 (a − b) 4 = a 4 − 4a 3b + 6a 2b 2 − 4ab3 + b 4 (a − b)5 = a 5 − 5a 4b + 10a 3b 2 − 10a 2b3 + 6ab 4 − b5 The coeficient is negative if the power exponent of b is odd number Practice Exercises 1. ( x + 3)3 2. ( x − 2) 4 3. ( x − 1)5 4. ( x + 1)6 23
  • 24. 3. Division Case 1: If the devisor is monomial 4 x 3 − 8 x 2 + 6 x 4 x3 8x 2 6x = − + = 2x2 − 4x + 3 2x 2x 2x 2x Sometimes, we may have remainder 4 x 3 − 8 x 2 + 6 x + 3 4 x3 8x 2 6x 3 3 = − + + = 2x2 − 4x + 3 + 2x 2x 2x 2x 2x 2 x Case 2: If the devisor is binomial Use factoring ab + ac = a (b + c) or ab − ac = a (b − c ) 4x2 + 2x 2 x ×2 x + 2 x 2 x (2 x + 1) = = = 2x 2x +1 2x +1 2x +1 2 4x2 − 6x − 3 4x2 + 2x − 8x − 4 + 1 ( 4 x + 2x ) − ( 8x + 4) + 1 = = 2x +1 2x +1 2x +1 4x2 + 2x 8x + 4 1 1 = − + = 2x − 4 + 2x +1 2x +1 2x +1 2x +1 24
  • 25. Other methods to get result 4x2 − 6x − 3 1 = 2x − 4 + 2x +1 2x +1 vertical division 25
  • 26. 13 2 4m − 8m + 4m + 6 1 2 = 2m − 3m + + 2m − 1 2 2m − 1 vertical way with number only 3 2 26
  • 27. 3x 3 − 2 x 2 − 150 3 x 3 − 2 x 2 + 0 ×x − 150 12 x − 158 = = 3x − 2 + 2 2 x −4 x + 0 ×x − 4 x2 − 4 Put 0s for the missing terms Remainder 27
  • 28. Exercises Do divisions −4 x 7 − 14 x 6 + 10 x 4 − 14 x 2 1. −2 x 2 10 x8 − 16 x 6 − 4 x 4 2. −2 x 6 Use vertical devision with number only 12 x 3 − 2 x + 5 3. x −3 6 x 4 + 9 x3 + 2 x 2 − 8 x + 7 4. 3x 2 − 2 5x4 + 2x2 − 3 5. x2 − x + 1 28
  • 29. Factoring Factoring is the reverse of polynomial multiplication and based on ab + ac = a (b + c ) here a could number or formula Factor the Greatest Common Factor GCF, including the largest posssible common number factor and lowest power of x or anything Examples: 9 x 2 + 6 x −12 = (3 × x 2 + 3 × x − 3 × ) = 3(3 x 2 + 2 x − 4) 3 2 4 9 x 5 + 6 x 3 − 12 x 2 = (3 x 2 × x 3 + 3 x 2 × x − 3 x 2 × ) = 3 x 2 (3 x 3 + 2 x − 4) 3 2 4 6 x 2t + 8 xt + 12t = (2t × x 2 + 2t × x + 2t × ) = 2t (3 x 2 + 4 x + 6) 3 4 6 14(m + 1)3 − 28(m + 1) 2 − 7( m + 1) = ( 7(m + 1) × m + 1) 2 − 7(m + 1) × m + 1) − 7( m + 1) × ) 2( 4( 1 = 7(m + 1) ( 2(m + 1) 2 − 4(m + 1) − 1) 29
  • 30. Group Factoring If there are four terms, we can group the 1st two and the last two terms. Then do the preliminary factors on two grous and factor again. x 3 + 2 x 2 + 2 x + 4 = ( x 3 + 2 x 2 ) + ( 2 x + 4 ) = x 2 ( x + 2 ) + 2( x + 2) = ( x 2 ( x + 2 ) + 2( x + 2) ) = ( x + 2 ) ( x 2 + 2 ) 4 x 3 + 2 x 2 − 6 x − 3 = ( 4 x 3 + 2 x 2 ) − ( 6 x + 3 ) = 2 x 2 ( 2 x + 1) − 3(2 x + 1) = ( 2 x 2 ( 2 x + 1) − 3(2 x + 1) ) = ( 2 x + 1) ( 2 x 2 − 3) mp 2 + 7m 2 + 3 p 2 + 21m = ( mp 2 + 7m 2 ) + ( 3 p 2 + 21m ) = m ( p 2 + 7m ) + 3 ( p 2 + 7m ) ( ) = m ( p 2 + 7m ) + 3 ( p 2 + 7 m ) ¬ − − We can skip this = ( p 2 + 7 m ) ( m + 3) 30
  • 31. Factor the following Exercises 1. 12m + 60 2. 4 p 3 q 4 − 6 p 2 q 5 3. 4k 2 m3 + 8k 4 m3 − 12k 2 m 4 4. 4( y − 2) 2 + 3( y − 2) 5. 6 st + 9t − 10 s − 15 6. 20 z 2 − 8 x + 5 pz 2 − 2 px 31
  • 32. Quadratic polynomials Factoring is the reverse of polynomial multiplication. (3x − 4)(2 x + 5) = 6 x 2 + 7 x − 20 is multiplication 6 x 2 + 7 x − 20 = (3x − 4)(2 x + 5) is factoring How to obtain numbers 3, − 4, 2 and 5 from 6, 7 and − 20? Because 6 x 2 = 3 x ×2 x so we have 6 = 3 ×2 and − 20= ( −4 ) × 5 Also 7 x = 3 x × + ( −4 ) ×2 x so we have 7 = 3 × + ( −4 ) ×2 5 5 Therefore we have chcart (answer are 4 corner numbers) 2x+5 3x−4 32
  • 33. Example 2. Factor 6 x 2 − 13 x + 6 This is not match This is match 2x−3 3x−2 Answer: 6 x 2 − 11x + 6 = (2 x − 3)(3 x − 2) 33
  • 34. Example 3. Factor 4 x 2 − 11xy + 6 y 2 4x−3y x−2y Answer: 4 x 2 − 11xy + 6 y 2 = (4 x − 3 y )( x − 2 y ) Example 4. Factor 6 p 2 − 7 p − 5 Answer: 6 p 2 − 7 p − 5 = (2 p + 1)(3 p − 5) 34
  • 35. Example 5. Factor x − 11x + 30 2 Answer: x 2 − 11x + 30 = ( x − 5)( x − 6) Example 6. Factor a 2 − 5ab − 14b 2 Answer: a 2 − 5ab − 14b 2 = (a + 2b)(a − 7b) 35
  • 36. Note: If the first coefficient is one like x 2 + px + q then we only need to decompose q to the product of two number such that their sum is p. Examples: 1. x 2 − 11x + 30 because 30 = ( −5)(−6) and so 2. 3. x 2 − 11x + 30 = ( x − 5)( x − 6) a 2 − 5a − 14 because − 14 = 2 ×(−7) and so (−5) + (−6) = −11 2 + (−7) = −5 a 2 − 5a − 14 = (a + 2)(a − 7) x 2 + 10 x − 39 because − 39 = 13 ×( −3) and 13 + ( −3) = 10 so x 2 + 10 x − 39 = ( x + 13)( x − 3) 36
  • 37. Factor the following Exercises 1. 8h 2 − 2h − 21 2. 3m 2 + 14m + 8 3. 9 y 2 − 18 y + 8 4. 6k 2 − 5kp − 6 p 2 5. 5a 2 − 7 ab − 6b 2 6. 24a 4 + 10a 3b − 2a 2b 2 7. 18 x 5 + 15 x 4 z − 75 x 3 z 2 8. x 2 + 12 x + 27 9. x 2 + x − 12 10. x 2 + 11x − 12 11. x 2 + 10 x − 24 12. x 2 − 5 x − 24 13. x2 − 2 x + 5 37
  • 38. Prime Polynomial If a integer coefficients polynomial cannot be factored to a product of polynomials with integer, then it called prime polynomials. 1. Suppose that m and n are positive integers, then mx 2 + n is prime such as x 2 + 9, 2 y 2 + 5, x2 + y2 are all prime. 2. For quadratice polynoimal ax 2 + bx + c if number b 2 − 4ac < 0 Example: x 2 + x + 1, then ax 2 + bx + c is prime x 2 + 2 xy + 3 y 2 are all prime. if number b 2 − 4ac is not a square number then ax 2 + bx + c is prime Example: x 2 + 3 x + 1, b 2 − 4ac = 32 − 4 × × = 5 not a square number 11 so x 2 + 3 x + 1 is prime 38
  • 39. Use Formulas We have the following formulas x 2 − y 2 = ( x + y )( x − y ) difference of squares x 2 + 2 xy + y 2 = ( x + y ) 2 perfect saqure of sum x 2 − 2 xy + y 2 = ( x − y ) 2 perfect saqure of difference x 3 + y 3 = ( x + y )( x 2 − xy + y 2 ) sum of cubic powers x 3 − y 3 = ( x − y )( x 2 + xy + y 2 ) difference of cubic powers Examples 1. 4m 2 − 9 = (2m) 2 − 32 = (2m + 3)(2m − 3) 1 3 1 3 2 2 2 2 x y x+ y x− y 2. 4 x 2 − 9 y 2 = (2 x) 2 − (3 y ) 2 = (2 x + 3 y )(2 x − 3 y ) 39
  • 40. 3. 256k − 81m = ( 16k 4 4 = ( 16k 2 + 9m 2 ) − ( 9m ) = ( 16k ) ( (4k ) − (3m) ) 2 2 2 2 2 2 + 9m 2 ) ( 16k 2 − 9m 2 ) 2 = ( 16k 2 + 9m 2 ) ( 4k + 3m ) ( 4k − 3m ) 4. (a + 2b) 2 − 4c 2 = ( a + 2b) 2 − (2c) 2 = [ ( a + 2b) + 2c ] [ ( a + 2b) − 2c ] = (a + 2b + 2c)(a + 2b − 2c) 5. x 2 + 2 x + 1 = ( x + 1) 2 6. x 2 + 2 xy + y 2 = ( x + y ) 2 7. x 2 − 6 x + 9 = ( x − 3) 2 8. 25 y 2 − 10 y + 1 = ( 5 y ) − 2 × y × + 12 = (5 y − 1) 2 5 1 2 9. 4m + 28m + 49 = ( 2m ) + 2 ×2m ×7 + (7) 2 = (2m + 7) 2 2 2 40
  • 41. 10. x − 6 x + 9 − y = ( x − 3) − ( y ) = ( x −3+ y ) ( x −3− y ) 11. ( m + 14m + 49 ) − ( y − 10 y + 25 ) = (m + 7) − ( y − 5) 2 4 2 2 2 2 2 2 2 2 2 = [ (m + 7) + ( y − 5) ] [ (m + 7) − ( y − 5) ] = (m + y − 2)(m − y + 12) 12. x 3 + 27 = x 3 + 33 = ( x + 3)( x 2 − x × + 32 ) = ( x + 3)( x 2 − 3 x + 9) 3 13. m − 64n = m − ( 4n ) = ( m − 4n)  m 2 + m ×4n + ( 4n) 2    3 3 3 3 = (m − 4n) ( m 2 + 4mn + 16n 2 ) 14. 8q + 125 p = ( 2q 6 9 = (2q 2 = (2q 2 ) +(5p ) + 5 p ) ( 2q ) − 2q × p + (5 p ) 5   + 5 p ) ( 4q − 10q p + 25 p ) 2 3 3 3 2 2 3 3 4 2 2 3 3 3 2    6 41
  • 42. Exercises Factor the following by formulas 1. 9m 2 − 12m + 4 2. 16 p 2 + 40 p + 25 3. 36 x 2 − 60 xy + 25 y 2 4. 9 x 2 − 6 x3 + x 4 5. 4 x 2 y 2 + 28 xy + 49 6. ( a − 2b) 2 − 6( a − 2b) + 9 7. 9m 2 n 2 − 4 p 2 8. ( a + 2b) 2 − 25( a − 3b) 2 9. 8 x3 + 27 10. x 4 − 81 11. 27 − ( m + 2n)3 12. x 4 − 16 13. x4 − 5x2 + 4 42

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