Simplify (3x-2)^3 - (4x+5)^2
Can you show it step-by-step please
You need the Binomial Theorem:
(a+b)^n = a^n + n a^(n-1) b + ...
So, you have
(3x-2)^3
= (3x)^3 - 3(3x)^2(2) + 3(3x)(2^2) - 2^3
= 27x^3 - 54x^2 + 36x - 8
(4x+5)^2
= (4x)^2 + 2(4x)(5) + 5^2
= 16x^2 + 40x + 25
Subtract them to get
27x^3 - 70x^2 - 4x - 33
Of course! To simplify the expression (3x-2)^3 - (4x+5)^2, we can follow these steps:
Step 1: Expand (3x-2)^3
To expand (3x-2)^3, we can use the binomial expansion formula, which states that (a+b)^n = a^n + n.a^(n-1).b + n.(n-1)/2!.a^(n-2).b^2 + ... + b^n.
In this case, a = 3x and b = -2. Plugging those values into the formula, we get:
(3x-2)^3 = (3x)^3 + 3.(3x)^2.(-2) + 3.(3x).(-2)^2 + (-2)^3
= 27x^3 - 54x^2 + 36x - 8
Step 2: Expand (4x+5)^2
Similarly, we can expand (4x+5)^2 using the same binomial expansion formula:
(4x+5)^2 = (4x)^2 + 2.(4x).(5) + 5^2
= 16x^2 + 40x + 25
Step 3: Substitute the expanded expressions into the original expression
Now that we have expanded both (3x-2)^3 and (4x+5)^2, we can substitute them back into the original expression:
(3x-2)^3 - (4x+5)^2
= (27x^3 - 54x^2 + 36x - 8) - (16x^2 + 40x + 25)
Step 4: Simplify by combining like terms
To simplify further, we can combine like terms:
= 27x^3 - 54x^2 + 36x - 8 - 16x^2 - 40x - 25
= 27x^3 - (54x^2 + 16x^2) + (36x - 40x) - (8 + 25)
= 27x^3 - 70x^2 - 4x - 33
Therefore, the simplified expression is 27x^3 - 70x^2 - 4x - 33.