Subtract. Simplify by removing a factor of 1 when possible.
(10bf)/(b^2-f^2)-(b-f)/b+f)
Thank you for your help
the actual subtraction part is easy: it's two factions being subtracted, which can only be done if they have the SAME denominator. So... the common denominator for the new fraction will be
(b^2 - f^2)(b + f)
the numerator is
(10bf)*(b + f) - (b - f)*(b^2 - f^2)
the new fraction is
(10bf)*(b + f) - (b - f)*(b^2 - f^2)
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(b^2 - f^2)(b + f)
To subtract and simplify the expression (10bf)/(b^2-f^2)-(b-f)/(b+f), follow these steps:
Step 1: Factor the numerator and denominator of the first fraction.
(10bf)/(b^2-f^2) = (10bf)/((b+f)(b-f))
Step 2: Bring the second fraction to a common denominator with the first fraction.
(b-f)/(b+f) = -((f-b)/(b+f))
Step 3: Rewrite the expression with a common denominator for both fractions.
(10bf)/((b+f)(b-f)) - ((f-b)/(b+f))
Step 4: Simplify the expression by multiplying the numerators and denominators.
(10bf - (f-b))/(b^2 - f^2)
Step 5: Combine like terms in the numerator.
(10bf - f + b)/(b^2 - f^2)
Thus, the simplified form of the expression is:
(10bf - f + b)/(b^2 - f^2)
To simplify the given expression, let's break it down step by step.
Step 1: Factor the numerator and the denominator
The numerator contains "10bf" which cannot be factored further.
The denominator contains two terms: (b^2-f^2) and (b-f). We can further simplify (b^2-f^2) using the difference of squares identity, which states that a^2 - b^2 = (a + b)(a - b). Applying this identity, we get:
(b^2 - f^2) = (b + f)(b - f).
Now we have:
(10bf)/((b + f)(b - f)) - (b - f)/(b + f).
Step 2: Find the common denominator
To subtract the fractions, we need to find a common denominator for the two terms. The common denominator is (b + f)(b - f).
Step 3: Rewrite the fractions with the common denominator
To rewrite the fractions with the common denominator, we need to multiply the numerator and denominator of each fraction by the missing factor.
For the first fraction:
(10bf)/((b + f)(b - f)) --> (10bf*(b + f))/((b + f)(b - f)).
For the second fraction:
(b - f)/(b + f) --> ((b - f)*(b - f))/((b + f)(b - f)).
Note that we multiplied the numerator and denominator of the second fraction by (b - f) to make the denominators match.
Now we have:
(10bf*(b + f))/((b + f)(b - f)) - ((b - f)*(b - f))/((b + f)(b - f)).
Step 4: Combine the fractions
The two fractions share the same denominator, so we can now combine them:
(10bf*(b + f) - (b - f)*(b - f))/((b + f)(b - f)).
Step 5: Simplify the numerator
Now, we simplify the numerator.
For the first term (10bf*(b + f)):
Combine the terms using distribution:
10bf*(b + f) = 10bf*b + 10bf*f = 10b^2f + 10bf^2.
For the second term ((b - f)*(b - f)):
Use the distributive property again:
(b - f)*(b - f) = (b^2 - 2bf + f^2).
Now, the expression becomes:
(10b^2f + 10bf^2 - (b^2 - 2bf + f^2))/((b + f)(b - f)).
Step 6: Simplify further
Now we can simplify the expression by combining like terms in the numerator:
10b^2f + 10bf^2 - (b^2 - 2bf + f^2) = 10b^2f + 10bf^2 - b^2 + 2bf - f^2.
Rearranging the terms:
(10b^2f - b^2) + (10bf^2 + 2bf) - f^2.
Step 7: Factor out common factors
We can factor out common factors from each group:
b^2(10f - 1) + bf(10f + 2) - f^2.
Now, we have the simplified expression:
(b^2(10f - 1) + bf(10f + 2) - f^2)/((b + f)(b - f)).
And that's the simplified form of the given expression after subtracting and removing a factor of 1 when possible.