I am having trouble with this problem.
perform the indicated operations and simpily the result as much as possible
a) (3x - 2) (x^2 + 2x - 4 )
the answer i got so far is 3x^3 + 4x^2 -16x + 8. Am i done or i have to conitnue to simplify? if so, what do i need to do?
b) 10x^3y^2 - 14x^2y^2 - 12y^2
the answer i got is 2y^2 ( 5x^3 - 7x^2 - 12 )
once again i am having trouble simplify this problem. Please explain
C) ( x - 1 ) ^ 7/2 - ( x - 1 ) ^ 3/2
i don't know what to do with this problem. Please explain.
Thank you.
The answers a) and b) are correct.
c )
( x - 1 ) ^ ( 7 / 2 ) - ( x - 1 ) ^ ( 3 / 2 ) =
( x - 1 ) ^ ( 3 / 2 ) * [ ( x - 1 ) ^ ( 4 / 2 ) - 1 ] =
( x - 1 ) ^ ( 3 / 2 ) * [ ( x - 1 ) ^ 2 - 1 ] =
( x - 1 ) ^ ( 3 / 2 ) * [ x ^ 2 - 2x + 1 - 1 ] =
( x - 1 ) ^ ( 3 / 2 ) * ( x ^ 2 - 2x ) =
( x - 1 ) ^ ( 3 / 2 ) * x * ( x - 2 )
a) To simplify the expression (3x - 2)(x^2 + 2x - 4), you need to distribute each term in the first set of parentheses to each term in the second set of parentheses. This means you multiply each term in the first set with each term in the second set and then combine like terms.
(3x - 2)(x^2 + 2x - 4) = 3x * x^2 + 3x * 2x - 3x * 4 - 2 * x^2 - 2 * 2x + 2 * 4
Now, we can simplify this expression by combining like terms:
= 3x^3 + 6x^2 - 12x - 2x^2 - 4x + 8
= 3x^3 + (6x^2 - 2x^2) + (-12x - 4x) + 8
= 3x^3 + 4x^2 - 16x + 8
So, your answer is correct. You have simplified it as much as possible.
b) Let's simplify the expression 10x^3y^2 - 14x^2y^2 - 12y^2.
In this case, notice that all terms have a common factor of y^2. We can factor out y^2:
= y^2 (10x^3 - 14x^2 - 12)
Now, let's factor out the greatest common factor of the remaining terms, which is 2:
= 2y^2 (5x^3 - 7x^2 - 6)
So, you have simplified it correctly. The expression cannot be simplified further as there are no further common factors or like terms to combine.
c) In order to simplify (x - 1)^(7/2) - (x - 1)^(3/2), we need to use exponent rules.
Let's rewrite the expression as a difference of squares:
= [(x - 1)^(7/2)]^2 - [(x - 1)^(3/2)]^2
Now, let's simplify the exponents of each term:
= (x - 1)^7 - (x - 1)^3
Now, we have two terms with the same base (x - 1), so we can combine them:
= [(x - 1)^7 - (x - 1)^3]
= [(x - 1)^3][(x - 1)^4 - 1]
= (x - 1)^3[(x - 1)(x - 1)^3 - 1]
= (x - 1)^3[(x^3 - 3x^2 + 3x - 1) - 1]
= (x - 1)^3(x^3 - 3x^2 + 3x - 1 - 1)
= (x - 1)^3(x^3 - 3x^2 + 3x - 2)
So, the simplified form of the expression is (x - 1)^3(x^3 - 3x^2 + 3x - 2).