a) 5(1/5x - 1)(5x - 1/5)
b)(2a - 3)^2 - (3a + 4)(3a - 4) +2(3a + 1)^2
c)(x/3 - 2)(x/2 - 3)
d) -1/5x(25x - 10y) + 1/3(12x-3y)
e) 3(x-2y)^3
well, we do not need the distributive property if you know FOIL
I do not dare to do a) because I do not know if you mean
(1/5)x or 1/(5x) when you write 1/5x
b)
(2a-3)(2a-3) = 4 a^2 - 12 a + 9
-(3a+4)(3a-4)= -9 a^2 + 0 a +16
2(3a+1)(3a+1)= 18a^2 + 12 a + 2
add
13 a^2 + 27
etc
Now the only trinomial is at e)
(x-2y)^2 = x^2 - 4 x y + 4 y^2
now we multiply that by (3x-6y)
use distributive property
(a+b)(c+d+e)
= a(c+d+e) + b(c+d+e)
so we have
+(3x)(x^2 - 4 x y + 4 y^2)
-(6y)(x^2 - 4 x y + 4 y^2)
=
+3 x^3 - 12 x^2y + 12 xy^2
-0 x^3 - 6 x^2 y + 24 xy^2 -24y^3
---------------------------------
=3x^2 -18 x^2y + 36 xy^2 -24 y^3
get it?
a) 5(1/5x - 1)(5x - 1/5)
To simplify this expression, we can use the distributive property and multiply each term inside the parentheses by the constant 5:
5 * 1/5x = (5/1) * (1/5) * x = x/1 = x
5 * (-1) = -5
Similarly:
(5x) * (1/5) = (5/1) * (1/5) * x = x/1 = x
(5x) * (-1/5) = (5/1) * (-1/5) * x = -x/1 = -x
The simplified expression is:
x * (x - 1) - 5 * (-1)
Simplified further:
x^2 - x + 5
b) (2a - 3)^2 - (3a + 4)(3a - 4) +2(3a + 1)^2
Let's work it step by step.
First, we can find the square of (2a - 3):
(2a - 3)^2 = (2a)^2 - 2 * 2a * 3 + (3)^2 = 4a^2 - 12a + 9
Next, we can expand (3a + 4)(3a - 4) using the FOIL method:
(3a + 4)(3a - 4) = 3a * 3a - 3a * 4 + 4 * 3a - 4 * 4 = 9a^2 - 12a + 12a - 16 = 9a^2 - 16
Then, we can find the square of (3a + 1):
(3a + 1)^2 = (3a)^2 + 2 * 3a * 1 + (1)^2 = 9a^2 + 6a + 1
Now, we can substitute these simplified expressions back into the original expression and simplify further:
(4a^2 - 12a + 9) - (9a^2 - 16) + 2(9a^2 + 6a + 1)
Simplified further:
4a^2 - 12a + 9 - 9a^2 + 16 + 18a^2 + 12a + 2
Combine like terms:
(4a^2 - 9a^2 + 18a^2) + (-12a + 12a) + (9 + 16 + 2)
Simplified further:
13a^2 + 27
c) (x/3 - 2)(x/2 - 3)
To simplify this expression, we can use the distributive property and multiply each term inside the parentheses:
(x/3) * (x/2) = (x * x) / (3 * 2) = x^2 / 6
(x/3) * (-3) = (x * -3) / 3 = -x
(-2) * (x/2) = (-2 * x) / 2 = -x
(-2) * (-3) = 6
The simplified expression is:
x^2 / 6 - x - x + 6
Combine like terms:
(x^2 / 6 - 2x - 2x + 6)
Simplified further:
x^2 / 6 - 4x + 6
d) -1/5x(25x - 10y) + 1/3(12x-3y)
To simplify this expression, we can use the distributive property and multiply each term inside the parentheses:
-1/5x * 25x = (-1/5 * 25 * x * x) = -5x^2
-1/5x * (-10y) = (-1/5 * -10 * x * y) = 2xy
1/3 * 12x = (1/3 * 12 * x) = 4x
1/3 * (-3y) = (1/3 * -3 * y) = -y
The simplified expression is:
-5x^2 + 2xy + 4x - y
e) 3(x-2y)^3
To simplify this expression, we need to apply the exponent to both the expression within the parentheses:
(x-2y)^3 = (x-2y)(x-2y)(x-2y)
We can use the distributive property to expand this expression. Start by multiplying the first two terms:
(x-2y)(x-2y) = x(x-2y) - 2y(x-2y)
Expand further:
x(x-2y) = x^2 - 2xy
-2y(x-2y) = -2xy + 4y^2
Now, multiply the expanded expression by the third term (x-2y):
(x^2 - 2xy)(x-2y) - (2xy - 4y^2)(x-2y)
Again, we can use the distributive property to expand:
(x^2 - 2xy)(x-2y) = x^3 -2x^2y - 2xy(x-2y) + 4xy^2
(2xy - 4y^2)(x-2y) = 2xy(x-2y) - 4y^2(x-2y) = 2xy^2 - 4y^3 - 4xy(x-2y) + 8y^2(x-2y)
Now, substitute the expanded expressions back into the original expression and simplify further:
3(x-2y)^3 = 3(x^3 -2x^2y - 2xy(x-2y) + 4xy^2) - 3(2xy^2 - 4y^3 - 4xy(x-2y) + 8y^2(x-2y))
Simplified further:
3x^3 - 6x^2y - 6xy(x-2y) + 12xy^2 - 6(2xy^2 - 4y^3 - 4xy(x-2y) + 8y^2(x-2y))
Combine like terms and simplify further if needed.