(3x+2)^3(3x^2-1)^2 pls i need good explanation its important
To simplify the expression (3x+2)^3(3x^2-1)^2, we can first expand both the cube and the square.
Let's start with the first term, (3x+2)^3. To expand this expression, we can apply the binomial expansion formula, which states that for any binomial in the form (a + b)^n, the expanded form can be obtained by multiplying each term of the first binomial by each term of the second binomial, with the appropriate powers.
(3x+2)^3 = (3x+2)(3x+2)(3x+2)
Now, let's multiply each term of the first binomial by every term of the second binomial:
(3x+2)(3x+2) = 9x^2 + 12x + 4
Multiplying the second term (3x+2) by the third term (3x+2):
(9x^2 + 12x + 4)(3x+2) = 27x^3 + 36x^2 + 12x + 18x^2 + 24x + 8
Combining like terms, we get:
27x^3 + 54x^2 + 36x + 8
Now, let's move on to the second term, (3x^2-1)^2. To expand this expression, we apply the same binomial expansion formula:
(3x^2-1)^2 = (3x^2-1)(3x^2-1)
Multiplying each term of the first binomial by every term of the second binomial:
(3x^2-1)(3x^2-1) = 9x^4 - 3x^2 - 3x^2 + 1
Combining like terms, we get:
9x^4 - 6x^2 + 1
Now, we multiply the expanded forms of (3x+2)^3 and (3x^2-1)^2 together:
(27x^3 + 54x^2 + 36x + 8)(9x^4 - 6x^2 + 1)
Using the distributive property, we multiply each term of the first expression by each term of the second expression:
27x^3 * 9x^4 + 27x^3 * (-6x^2) + 27x^3 * 1 +
54x^2 * 9x^4 + 54x^2 * (-6x^2) + 54x^2 * 1 +
36x * 9x^4 + 36x * (-6x^2) + 36x * 1 +
8 * 9x^4 + 8 * (-6x^2) + 8 * 1
Simplifying each term, we get:
243x^7 - 162x^5 + 27x^3 +
486x^6 - 324x^4 + 54x^2 +
324x^5 - 216x^3 + 36x +
72x^4 - 48x^2 + 8
Combining like terms and arranging in descending order of exponents:
243x^7 + 486x^6 - 162x^5 + 324x^5 + 72x^4 - 324x^4 - 216x^3 + 27x^3 + 36x -
48x^2 + 54x^2 + 8
Simplifying further:
243x^7 + 810x^6 + 162x^5 - 648x^4 - 189x^3 + 90x^2 + 36x + 8
Therefore, the simplified form of (3x+2)^3(3x^2-1)^2 is 243x^7 + 810x^6 + 162x^5 - 648x^4 - 189x^3 + 90x^2 + 36x + 8.