How would I simplify the following, but leave it in factored form?
[(2x+7)^6 x 5(x-2)^4] + [(x-2)^5 x 6(2x+7)^5 x 2]
If you can also explain how to do it step by step, that would be VERY appreciated. THANK YOU SO MUCH.
I changed the variable to y to avoid confusion with the x used for multiplication.
[(2y+7)^6 x 5(y-2)^4] + [(y-2)^5 x 6(2y+7)^5 x 2]
(2y+7)^5 and (y-2)^4 are common to both terms.
bring the common factors outside:
=((2y+7)^5)x((y-2)^4)x[(2y+7)+(y-2)x2]
There are 2 terms in the rightmost factor. Combine like items.
Oops!
An error on my part!
The last line should be:
=((2y+7)^5)x((y-2)^4)x[5x(2y+7)+6x(y-2)x2]
To simplify the given expression while leaving it in factored form, we can follow these steps:
Step 1: Identify common factors
In the expression [(2x+7)^6 x 5(x-2)^4] + [(x-2)^5 x 6(2x+7)^5 x 2], we have common factors in both terms: (2x+7) and (x-2).
Step 2: Group the common factors
We can group the common factors together. So, the expression can be rewritten as follows:
[(2x+7)^6 x 5(x-2)^4] + [(2x+7)^5 x (x-2)^5 x 6 x 2]
Step 3: Combine the common factors
Now, we can combine the common factors of (2x+7) and (x-2). Rewrite the expression as:
(2x+7)^5 x (x-2)^4 [(2x+7) x 5(x-2)] + [(2x+7)^5 x (x-2)^5 x 6 x 2]
Step 4: Simplify the coefficients
Calculate the product of the coefficients within the square brackets and the product outside:
(2x+7)^5 x (x-2)^4 [10(x-2) + 12(2x+7)]
Step 5: Expand and simplify
Distribute the common factors (2x+7)^5 and (x-2)^4 into the square brackets:
(2x+7)^5 x (x-2)^4 [10x - 20 + 24x + 84]
Step 6: Combine like terms
Combine the like terms within the square brackets:
(2x+7)^5 x (x-2)^4 [34x + 64]
Finally, we have simplified the expression to:
(2x+7)^5 x (x-2)^4 [34x + 64]
That's the final expression in factored form after simplification.