AlgebrAic factorization
1)6(2a-3)(a+7)+18(2a-3)(3a+2)
2) (3x-4)(5x+1)-(5x+3)(4-3x)
I know number 1 and 2 but just wanted to make sure if my answers
I need help in this only:
3) (3x-1)^2 + 5(1-3x)+3x(1-3x)^2
3. 9x^2-6x+1 + 5-15x + 3x(9x^2-6x+1)
9x^2-6x+1 + 5-15x + 27x^3-18x^2+3x
27x^3-9x^2-18x+6
3(9x^3-3x^2-6x+2)
To simplify algebraic expressions like the ones you mentioned, you can follow these steps:
1) Distribute any constants or coefficients that are outside of the parentheses to each term inside the parentheses.
2) Apply the distributive property by multiplying each term inside the first set of parentheses by each term inside the second set of parentheses.
3) Combine like terms.
Let's go through each expression:
1) 6(2a-3)(a+7) + 18(2a-3)(3a+2)
Step 1: Distribute the coefficients to each term inside the parentheses:
6 * 2a = 12a
6 * -3 = -18
18 * 2a = 36a
18 * -3 = -54
Now the expression becomes:
12a(a+7) - 18(a+7) + 36a(3a+2) - 54(3a+2)
Step 2: Apply the distributive property:
12a * a = 12a^2
12a * 7 = 84a
-18 * a = -18a
-18 * 7 = -126
36a * 3a = 108a^2
36a * 2 = 72a
-54 * 3a = -162a
-54 * 2 = -108
Now the expression becomes:
12a^2 + 84a - 18a - 126 + 108a^2 + 72a - 162a - 108
Step 3: Combine like terms:
12a^2 + 108a^2 + 84a - 18a + 72a - 162a - 126 - 108
120a^2 - 204a - 234
So, the simplified expression is 120a^2 - 204a - 234.
2) (3x-4)(5x+1) - (5x+3)(4-3x)
Step 1: Distribute the coefficients to each term inside the parentheses:
3x * 5x = 15x^2
3x * 1 = 3x
-4 * 5x = -20x
-4 * 1 = -4
5x * 4 = 20x
5x * -3x = -15x^2
1 * 4 = 4
1 * -3x = -3x
Now the expression becomes:
15x^2 + 3x - 20x - 4 - 15x^2 + 20x + 4 - 3x
Step 3: Combine like terms:
(15x^2 - 15x^2) + (3x - 20x + 20x - 3x) + (-4 + 4)
0 + 0 + 0
So, the simplified expression is 0.
3) (3x-1)^2 + 5(1-3x) + 3x(1-3x)^2
To simplify this expression, we'll follow the same steps:
Step 1: Simplify the squared term:
(3x-1)^2 = (3x-1)(3x-1) = 9x^2 - 6x + 1
Step 2: Distribute the constant and coefficients:
5 * 1 = 5
5 * -3x = -15x
3x * 1 = 3x
3x * (1-3x) = 3x - 9x^2
Now the expression becomes:
9x^2 - 6x + 1 + 5 - 15x + 3x - 9x^2
Step 3: Combine like terms:
(9x^2 - 9x^2) + (-6x + 3x - 15x) + (1 + 5)
0 + -18x + 6
So, the simplified expression is -18x + 6.
I hope this helps you verify your answers!