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UNIT 1  : ALGEBRA PREP

LESSON 4:  OPERATIONS WITH POLYNOMIALS HOMEWORK QUESTIONS

 

 

HOMEWORK QUESTIONS:

 

1. Expand and simplify:

a) (x2 – 3x + 4) + (4x2 – 2x + 5)                                                          b) (x2 – 2x – 3) – (5x2 + 3x – 2)

c) (-3x2 + 2x – 5) – 2x(3x – 4) + 3(x – 5)                                            d) 3x2(2x3 – 6x – 2) – x(5x3 – 3x2 + 1)

 

2. Expand and simplify:

a) 3[2x – (x – 5)] –2x[5x – (3x – 2)]                                                   b) (2x – 1)(3x2 – 5x – 2)

c) (x2 + 2)(x2 – 5x + 3)                                                                                    d) 5(3 – 2x)(4x – 1) – 2x(3x + 2)2

e) (2a – 3)2(a – 2) – 3a(a – 1)(4a + 2)                                                  f) (5a – 1)(2a3 – 3a2 + a – 5)

g) (2a – 3b + c)3

 

3. Expand and simplify:

                                                                        

 

 

4. Factor Fully:

 

 

Solutions:

 

1. Expand and simplify:

a) (x2 – 3x + 4) + (4x2 – 2x + 5)                                                          b) (x2 – 2x – 3) – (5x2 + 3x – 2)

c) (-3x2 + 2x – 5) – 2x(3x – 4) + 3(x – 5)                                            d) 3x2(2x3 – 6x – 2) – x(5x3 – 3x2 + 1)

Solutions:

a) (x2 – 3x + 4) + (4x2 – 2x + 5) = x2 – 3x + 4 + 4x2 – 2x + 5

                                                  = 5x2 – 5x + 9

 

b) (x2 – 2x – 3) – (5x2 + 3x – 2) = (x2 – 2x – 3) – 1(5x2 + 3x – 2)

                                                  = x2 – 2x – 3 – 5x2 – 3x + 2

                                                  = -4x2 – 5x – 1

 

c) (-3x2 + 2x – 5) – 2x(3x – 4) + 3(x – 5) = -3x2 + 2x – 5 – 6x2 + 8x + 3x – 15

                                                                 = -9x2 + 13x - 20

 

d) 3x2(2x3 – 6x – 2) – x(5x3 – 3x2 + 1) = 6x5 – 18x3 – 6x2 – 5x4 + 3x3 – x

                                                            = 6x5 – 5x4 – 15x3 – 6x2 – x

 

2. Expand and simplify:

a) 3[2x – (x – 5)] –2x[5x – (3x – 2)]                                                   b) (2x – 1)(3x2 – 5x – 2)

c) (x2 + 2)(x2 – 5x + 3)                                                                                    d) 5(3 – 2x)(4x – 1) – 2x(3x + 2)2

e) (2a – 3)2(a – 2) – 3a(a – 1)(4a + 2)                                                  f) (5a – 1)(2a3 – 3a2 + a – 5)

g) (2a – 3b + c)3

Solutions:

a) 3[2x – (x – 5)] –2x[5x – (3x – 2)] = 3(2x – x + 5) – 2x(5x – 3x + 2)

                                                            = 3(x + 5) – 2x(2x + 2)

                                                            = 3x + 15 –4x2 – 4x

                                                            = - 4x2 – x + 15

 

b) (2x – 1)(3x2 – 5x – 2) = 2x(3x2 – 5x – 2) – 1(3x2 – 5x – 2)

                                       = 6x3 – 10x2 – 4x – 3x2 + 5x + 2

                                       = 6x3 – 13x2  + x + 2

 

c) (x2 + 2)(x2 – 5x + 3)            = x2(x2 – 5x + 3) + 2(x2 – 5x + 3)

                                    = x4 – 5x3 + 3x2 + 2x2 – 10x + 6

                                    = x4 – 5x3 + 5x2 – 10x + 6

 

d) 5(3 – 2x)(4x – 1) – 2x(3x + 2)2 = 5(12x – 3 – 8x2 + 2x) – 2x(3x + 2)(3x + 2)

                                                     = 5(-8x2 + 14x – 3) – 2x(9x2 + 6x + 6x + 4)

                                                     = -40x2 + 70x – 15 – 18x3 – 12x2 – 12x2 – 8x

                                                     = - 18x3 – 64x2 + 62x – 15

 

e) (2a – 3)2(a – 2) – 3a(a – 1)(4a + 2)  = (4a2 – 12a + 9)(a – 2) – 3a(4a2 – 2a – 2)

                                                            = 4a3 – 8a2 –12a2 + 24a + 9a – 18 – 12a3 + 6a2 + 6a  

                                                            = -8a3 – 14a2 + 39a – 18

 

 f) (5a – 1)(2a3 – 3a2 + a – 5) = 5a(2a3 – 3a2 + a – 5) – 1(2a3 – 3a2 + a – 5)

                                                = 10a4 – 15a3 + 5a2 – 25a – 2a3 + 3a2 – a + 5

                                                = 10a4 – 17a3 + 8a2 – 26a + 5

 

g) (2a – 3b + c)3 = (2a – 3b + c) (2a – 3b + c) (2a – 3b + c)

                           = (2a - 3b + c)(4a2 – 6ab + 2ac –6ab +9b2 –3bc +2ac – 3bc +c2)

                     = (2a – 3b + c)(4a2 + 9b2 + c2 –12ab + 4ac – 6bc)

                           = 8a3 + 18ab2 +2ac2 –24a2b +8a2c –12abc –12a2b – 27b3 –3bc2 +36ab2 –12abc + 18b2c + 4a2c +9b2c +c3 – 12abc + 4ac2 –6bc2

                     = 8a3 – 27b3 + c3 + 54ab2 +6ac2 –36a2b + 12 a2c – 9bc2 + 27b2c –36abc

 

3. Expand and simplify:

                                                                        

 

 

4. Factor Fully:

 

 

           

 

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