Mathematics Warm Ups:

1. Solve the following: (x^{3} + 2x^{2}  5x + 7) + (4x^{3}  5x^{2} + 3).
Answer:
= x^{3} + 4x^{3} + 2x^{2}  5x^{2}  5x + 7 + 3
= (1 + 4)x^{3} + (2  5)x^{2}  5x + 10
= 5x^{3}  3x^{2}  5x + 10
2. Solve the following: (5x^{3} + 3x^{2}  3)  (x^{3}  4x^{2}  6x + 5)
Answer:
= 5x^{3} + 3x^{2}  3  x^{3} + 4x^{2} + 6x  5
= 5x^{3}  x^{3} + 3x^{2} + 4x^{2} + 6x  3  5
= (5  1)x^{3} + (3 + 4)x^{2} + 6x  8
= 4x^{3} + 7x^{2} + 6x  8
3. Solve the following: (4x  5)(2x^{2}  5x + 9)
Answer:
= 4x(2x^{2}  5x + 9)  5(2x^{2}  5x + 9)
= 8x^{3}  20x^{2} + 36x  10x^{2} + 25x  45
= 8x^{3} + (20  10)x^{2} + (36 + 25)x  45
= 8x^{3} 30x^{2} + 61x  45
4. Solve the following: (x^{2}  4x + 2)(4x^{3} + 3x + 2)
Answer:
= x^{2}(4x^{3} + 3x + 2)  4x(4x^{3} + 3x + 2) + 2(4x^{3} + 3x + 2)
= 4x^{5} + 3x^{3} + 2x^{2}  16x^{4}  12x^{2}  8x + 8x^{3} + 6x + 4
= 4x^{5}  16x^{4} + (3 + 8)x^{3} + (2  12)x^{2} + (8 + 6)x + 4
=4x^{5}  16x^{4} + 11x^{3}  10x^{2}  2x + 4
5. Solve the following: (x^{2} + xy + y^{2})(x  y)
Answer:
= x(x^{2} + xy + y^{2})  y(x^{2} + xy + y^{2})
= x^{3} + x^{2}y + xy^{2}  x^{2}y  xy^{2}  y^{3}
= x^{3}  y^{3}
