3.6. Solved problems

1. Let A = x² - 6x + 5, B = x³ - 4x² + 2, find the sum and difference of A, B and give degree of results.

   A + B = x² - 6x + 5 + x³ - 4x² + 2
         = x³ + x² - 4x² - 6x + 5 + 2
         = x³ - 3x² - 6x + 7
   degree = 3
   
   A - B =  x² - 6x + 5 - x³ + 4x² - 2
         = -x³ + x² + 4x² - 6x + 5 - 2
         = -x³ + 5x² - 6x + 3
   degree = 3
   

2. Multiply out (x + 1)(x - 2)(x + 3) and give degree of result.

   (x + 1)(x - 2)(x + 3) = (x² - 2x + x - 2)(x + 3) = 
   (x² - x - 2)(x + 3) = x³ + 3x² - x² - 3x - 2x - 6 = 
   x³ + 2x² - 5x - 6 
   degree = 3
   

3. Multiply out (x + y + z)(x + 2y + 3z) and give degree of result.

   (x + y + z)(x + 2y + 3z) = 
   x² + 2xy + 3xz + xy + 2y² + 3yz + xz + 2yz + 3z² = 
   x² + 2y² + 3z² + 3xy + 4xz + 5yz
   degree = 2
   

4. Multiply out (x + 1)⁴ and give degree of result.

   (x + 1)4 = (x² + 2x + 1)² = 
   x4 + 2x³ + x² + 2x³ + 4x² + 2x + x² + 2x + 1 = 
   x4 + 4x³ + 6x² + 4x + 1
   degree = 4
   

5. Divide 12x³ - 28x² + 17x - 3 by 3x - 1 and give quotient and its degree and remainder.

                   4x²  -  8x  + 3
   3x - 1 ) 12x³ - 28x² + 17x - 3
            12x³ -  4x² 
                 - 24x² + 17x - 3
                 - 24x² +  8x
                           9x - 3
                           9x - 3
                                0
   
   Quotient = 4x²  - 8x  + 3, degree = 2
   Remainder = 0
   

6. Divide x³ - 64 by x - 4 and give quotient and its degree and remainder.

                 x² +  4x + 16
   x - 4 ) x³ + 0x²  + 0x - 64
           x³ - 4x²
                4x² +  0x
                4x² - 16x
                      16x - 64
                      16x - 64
                             0
   
   Quotient = x² + 4x + 16, degree = 2
   Remainder = 0
   

7. Factor the polynomial x² - 2x + 6x.

   x² - 2x + 6x = x² + 4x = x(x + 4)
   

8. Factor the polynomial x² - 2.

   This is in the form of a difference of two squares.

   x² - 2 = (x - √2)(x + √2)
   

9. Factor the polynomial x³ - 8x² - 9x.

   x³ - 8x² - 9x = 
   x(x² - 8x - 9) = 		remove common factor
   x(x + 1)(x - 9) = 		reverse FOIL
   

10. Factor the polynomial 5x² + 13xy + 6y².

    Using the reverse FOIL method, get 5×1 = 5 and 3×2 = 6 which gives 13 = 5×2 + 3×1. Then,

    5x² + 13xy + 6y² = (x + 2y)(5x + 3y)
    

11. Factor the polynomial x⁴ + 4.

    Create a difference of two squares by adding and subtracting the term: 4x²

    x4 + 4 = 
    x4 + 4x² + 4 - 4x² = 	   add and subtract the term
    (x² + 2)² - 4x² =          difference of two squares
    (x² - 2x + 2)(x² + 2x + 2)