Help on Algebra 2 problems!?

Question

I need to make sure my answers are right so can anyone please help me solve these?
Write as a simplified polynomial:
1. (a+2)^2
2. (3a-2)(4a+3)
3. r^2s^2(4r-5s)
4. (5a-2) - (3-a)
5. (x^2-2)(x+5)
6. (-a)^2(2a^2)^3

Answer 1

1) Given polynomial is (a + 2)^2

It is in the form (a + b)^2 where a = a and b = 2

Therefore (a + 2)^2 = a^2 + 2^2 +2(a)(2)              [ Since (a + b)^2 = a^2 + b^2 + 2ab ]

                               = a^2 + 4 + 4a

                               = a^2 + 4a +4

Therefore (a + 2)^2 = a^2 + 4a +4

Answer 2

2) Given polinomial is (3a - 2) (4a + 3)

     This is in the form (a+b)(c+d) which is equal to a*c + a*d + b*c + b*d

     Therefore (3a - 2) (4a + 3) = (3a)(4a) + (3a)(3) - (2)(4a) - (2)(3)  

                                                = 12a^2 + 9a - 8a - 6                [ Since a*a = a^2 ]

                                                = 12a^2 +a -6

    Therefore (3a - 2) (4a + 3) = 12a^2 +a -6

Answer 3

3) Given polynomial is r^2s^2(4r - 5s)

This is in the form a(c - d) where a = r^2s^2, c = 4r and d = 5s

                                 = r^2s^2(4r) - r^2s^2(5s)   [ Since a(c - d) = ac - ad ]

                                 = 4r^3s^2 - 5r^2s^3            [ Since r^2*r =r^(2+1) = r^3 and s^2*s = s^3 ]

Therefore r^2s^2(4r - 5s) = 4r^3s^2 - 5r^2s^3    

Answer 4

4) Given polynomial is (5a-2) - (3-a)

                                 = (5a - 2) (-3 + a)

                                 = (5a -2) (a - 3)

This is in the form (a - b) (c - d) where a = 5a, b =2, c = a and d = 3

 Therefore (5a - 2)(a - 3) = 5a*a - 5a*3 - 2*a + 2*3   [ Since (a - b) (c - d) = ac - ad -bc + bd ]

                                        = 5a^2 -15a - 2a + 6         [ Since a*a =a^(1+1) = a^2 ]

                                        = 5a^2 -17a + 6  

Therefore (5a-2) - (3-a) = 5a^2 -17a + 6  

Answer 5

5) Given polynomial is (x^2 - 2) (x + 5)

    This is in the form ( a - b ) (c + d ) where a = x^2, b = 2, c = x and d = 5

    Therefore (x^2 - 2)(x + 5) = (x^2)(x) + (x^2)(5) - (2)(x) - (2)(5)  [Since (a-b)(c+d) = ac + ad - bc - bd]

                                            = x^3 + 5x^2 - 2x -10                       [ Since a^2*a = a^3 ]

    Therefore (x^2 - 2) (x + 5) = x^3 + 5x^2 - 2x -10

                                                

 

Answer 6

6) Given polynomial is (-a)^2 (2a^2)^3

                                 = (a^2) [ (2^3) (a^2)^3 ]               [ Since -*- = +, (ab)^3 = (a^3)(b^3) ]

                                 = a^2 ( 8 a^(2*3) )                       [ Since 2^3 = 2*2*2 = 8, (a^b)^c = a^(b*c) ]

                                 = a^2 ( 8 a^6 )

                                 = 8 a^2 a^6

                                 = 8 a^(2+6)                                 [ Since a^b * a^c = a^(b+c) ]

                                 = 8 a^8

   Therefore (-a)^2 (2a^2)^3 = 8 a^8