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UNIT 1 : PRECALCULUS 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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