2a) Simplify the expressions for A and B, where A=x+4/xsquared+9x+20 and B= 3xsquared-9x/xsquared+3x-18
b) Are the two expressions equivalent? Justify your answer.
c) Determine
i) A-B
ii) A•B
iii) B÷A
you probably meant:
A= (x+4)/(xsquared+9x+20)
= (x+4)/((x+4)(x+5))
= 1/(x+5) , x ≠ -4
B = (3xsquared-9x)/(xsquared+3x-18)
= 3x(x - 3)/((x+6)(x-3))
= 3x/(x+6) , x ≠ 3
clearly they are not equivalent
I am sure you can handle the algebra from this point.
To simplify the expressions for A and B, we need to factorize the given expressions as much as possible and then cancel out any common factors.
a) Simplifying expression A:
We have A = (x + 4) / (x^2 + 9x + 20)
First, we try to factorize the denominator (x^2 + 9x + 20):
We can rewrite the expression as (x + 4)(x + 5).
So, A becomes: A = (x + 4) / [(x + 4)(x + 5)]
Now, we can cancel out the common factor of (x + 4) in the numerator and denominator:
A = 1 / (x + 5)
Therefore, the simplified expression for A is A = 1 / (x + 5).
Similarly, let's simplify expression B:
We have B = (3x^2 - 9x) / (x^2 + 3x - 18)
First, we factorize the denominator (x^2 + 3x - 18):
We can rewrite the expression as (x + 6)(x - 3).
So, B becomes: B = (3x^2 - 9x) / [(x + 6)(x - 3)]
Now, we can cancel out the common factor of 3x in the numerator and denominator:
B = (x - 3) / (x + 6)
Therefore, the simplified expression for B is B = (x - 3) / (x + 6).
b) To determine if the two expressions A and B are equivalent, we need to compare them.
From the simplifications above, we have:
A = 1 / (x + 5)
B = (x - 3) / (x + 6)
The expressions A and B are not equivalent because their numerators and denominators are different.
c) Now, let's calculate the following:
i) A - B:
A - B = (1 / (x + 5)) - ((x - 3) / (x + 6))
To subtract fractions, we need to find a common denominator, which is (x + 5)(x + 6):
A - B = [(1 * (x + 6)) - ((x - 3) * (x + 5))] / [(x + 5)(x + 6)]
Expanding and simplifying, we get:
A - B = (x + 6 - (x^2 + 2x - 15)) / [(x + 5)(x + 6)]
Simplifying the numerator:
A - B = (-x^2 + 3x - 9) / [(x + 5)(x + 6)]
Therefore, A - B = (-x^2 + 3x - 9) / [(x + 5)(x + 6)].
ii) A • B:
A • B = [(x + 4) / (x + 5)] * [(x - 3) / (x + 6)]
To multiply fractions, we can multiply the numerators and denominators directly:
A • B = [(x + 4)(x - 3)] / [(x + 5)(x + 6)]
Expanding, we get:
A • B = (x^2 + x - 12) / [(x + 5)(x + 6)]
Therefore, A • B = (x^2 + x - 12) / [(x + 5)(x + 6)].
iii) B ÷ A:
B ÷ A = [(x - 3) / (x + 6)] ÷ [1 / (x + 5)]
To divide fractions, we can multiply the first fraction by the reciprocal of the second:
B ÷ A = [(x - 3) / (x + 6)] * [(x + 5) / 1]
Simplifying:
B ÷ A = (x - 3)(x + 5) / (x + 6)
Therefore, B ÷ A = (x - 3)(x + 5) / (x + 6).