15. -3x ( 5x + 1) (8 - 2x) =
-3x ( 5x * 8 + 1*8 - 2x*5x - 2x*1) =
-3x ( 40x + 8 - 10x^2 - 2x) =
-120x^2 - 24x + 30x^3 + 6x^2 =
30x^3 - 114x^2 - 24x
16. (a + b)(3a - b)(2a + 7b) =
(3a² - ab + 3ab - b²)(2a + 7b) =
(3a² + 2ab - b²)(2a + 7b) =
6a³ + 21a²b + 4a²b + 14ab² - 2ab² - 7b³ =
6a³ + 25a²b + 12ab² - 7b³
Explanation: Multiply any two terms first, in this case I multiplied (a+b) and (3a-b) first. Then with the result, multiply it with the last term.
17/18 follows the same procedure as 16.
19. (-42x¹⁰y⁵+12x⁸y³-6x²y) / (6x²y) =
[(-42x¹⁰y⁵) / (6x²y)] + [(12x⁸y³) / (6x²y)] + [(-6x²y) / (6x²y)]
(-7x⁸y⁴) + (2x⁶y²) + (-1)
-7x⁸y⁴ + 2x⁶y² - 1
Explanation: Separate the fraction into 3 fractions. Simplify using division and laws of exponents.
20 follows the same procedure as 19.
I got the set up part done but not sure how to do this >.<
[(16a^4/12a^3)]-[(40a^2/12a^3)]+[(24a/12a^3)]
aˣ/aʸ = aˣ⁻ʸ
For (16a⁴/12a³)], you can separate out the numbers and the variable so you get:
16/12 times a⁴/a³.
Think of 16/12 as a fraction and reduce it to its lowest terms --> 4/3
Same for a⁴/a³ --> a⁴⁻³ --> a
Put them back together and you get 4a/3
For [(40a^2/12a^3)], separate to 40/12 times a²/a³.
40/12 --> 10/3
a²/a³ --> a²⁻³ --> a⁻¹ --> 1/a
Put them together --> 10/3a
For [(24a/12a^3)], separate to 24/12 times a/a³.
24/12 --> 2
a/a³ --> a¹⁻³ --> a⁻² --> 1/a²
Put them together --> 2/a²
So, we get
(4a/3) - (10/3a) + (2/a²)
We can make it neater by keeping the denominators the same. The LCM is 3a².
Multiply 4a/3 by a²/a² to get 4a³/3a².
Multiply 10/3a by a/a to get 10a/3a².
Multiply 2/a² by 3/3 to get 6/3a².
Now, we can turn the expression into a single fraction.
(4a³/3a²) - (10a/3a²) + (6/3a²) --> (4a³-10a+6)/(3a²)
Alternatively, we can factor out the common term on the numerator.
(16a⁴-40a²+24a)/12a³
Factor out 8a:
8a(2a³-5a+3)/12a³
From here we can separate the fraction into 8a/12a³ times (2a³-5a+3)
8a/12a³ --> 2/3a²
Put them back together --> 2(2a³-5a+3)/3a²
Distribute the 2 --> (4a³-10a+6)/3a²
16
(a + b)(3a - b) (2a + 7b) expand the first two factors
( 3a *a + 3ab - ab - b^2) (2a + 7b) = simplify the terms in the left parentheses
(3a^2 + 2ab - b^2)(2a + 7b) = expand the first terms over the second
3a^2 * 2a + 2ab * 2a - b^2* 2a + 3a^2* 7b + 2ab*7b - b^2*7b =
6a^3 + 4a^2b - 2ab^2 + 21a^2b + 14ab^2 - 7b^3 =
6a^3 + 25a^2b + 12ab^2 - 7b^3