Simplify the following algebraic expressions: a2/(a−2b) + 4b2/(2b-a) 1/(b−1) − 1/(b+1) 1/(x−1) − 2/(x2-1)

Question

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here are the solutions to the problems:

1)

\frac{a^2}{a-2b} + \frac{4b^2}{2b-a}

Rewrite the denominator of the second term.
Note that $2b-a = -(a-2b)$ .
$\frac{a^2}{a-2b} + \frac{4b^2}{-(a-2b)} = \frac{a^2}{a-2b} - \frac{4b^2}{a-2b}$
Combine the fractions with the common denominator.
$\frac{a^2 - 4b^2}{a-2b}$
Factor the numerator using the difference of squares formula ( $x^2-y^2 = (x-y)(x+y)$ ).
$\frac{(a-2b)(a+2b)}{a-2b}$
Cancel out the common factor $(a-2b)$ .
$a+2b$
The simplified expression is $\boxed{a+2b}$ .

2)
$\frac{1}{b-1} - \frac{1}{b+1}$
Find a common denominator, which is $(b-1)(b+1)$ .
$\frac{1 \cdot (b+1)}{(b-1)(b+1)} - \frac{1 \cdot (b-1)}{(b+1)(b-1)}$
Perform the subtraction.
$\frac{(b+1) - (b-1)}{(b-1)(b+1)}$
Simplify the numerator and the denominator.
$\frac{b+1-b+1}{b^2-1} = \frac{2}{b^2-1}$
The simplified expression is $\boxed{\frac{2}{b^2-1}}$ .

3)
$\frac{1}{x-1} - \frac{2}{x^2-1}$
Factor the denominator of the second term using the difference of squares formula.
$x^2-1 = (x-1)(x+1)$ $\frac{1}{x-1} - \frac{2}{(x-1)(x+1)}$
Find a common denominator, which is $(x-1)(x+1)$ .
$\frac{1 \cdot (x+1)}{(x-1)(x+1)} - \frac{2}{(x-1)(x+1)}$
Perform the subtraction.
$\frac{(x+1) - 2}{(x-1)(x+1)}$
Simplify the numerator.
$\frac{x-1}{(x-1)(x+1)}$
Cancel out the common factor $(x-1)$ .
$\frac{1}{x+1}$
The simplified expression is $\boxed{\frac{1}{x+1}}$ .

4)
$\frac{a^3+1}{a^2-a+1} + \frac{a^3-1}{a^2+a+1}$
Factor the numerators using the sum and difference of cubes formulas
$x^3+y^3 = (x+y)(x^2-xy+y^2)$ and $x^3-y^3 = (x-y)(x^2+xy+y^2)$ .
$a^3+1 = (a+1)(a^2-a+1)$ $a^3-1 = (a-1)(a^2+a+1)$
Substitute the factored forms into the expression.
$\frac{(a+1)(a^2-a+1)}{a^2-a+1} + \frac{(a-1)(a^2+a+1)}{a^2+a+1}$
Cancel out common factors in each term.
$(a+1) + (a-1)$
Simplify the expression.
$a+1+a-1 = 2a$
The simplified expression is $\boxed{2a}$ .

5)
$\frac{m^2+mn+n^2}{m+n} + \frac{m^2-mn+n^2}{m-n}$
Find a common denominator, which is $(m+n)(m-n) = m^2-n^2$ .
$\frac{(m^2+mn+n^2)(m-n)}{(m+n)(m-n)} + \frac{(m^2-mn+n^2)(m+n)}{(m-n)(m+n)}$
Expand the numerators.
The first numerator is $(m^2+mn+n^2)(m-n) = m^3-n^3$ .
The second numerator is $(m^2-mn+n^2)(m+n) = m^3+n^3$ .
Substitute the expanded numerators back into the expression.
$\frac{m^3-n^3}{m^2-n^2} + \frac{m^3+n^3}{m^2-n^2}$
Combine the fractions.
$\frac{(m^3-n^3) + (m^3+n^3)}{m^2-n^2}$
Simplify the numerator.
$\frac{m^3-n^3+m^3+n^3}{m^2-n^2} = \frac{2m^3}{m^2-n^2}$
The simplified expression is $\boxed{\frac{2m^3}{m^2-n^2}}$ .

Simplify the following algebraic expressions: a2/(a-2b) + 4b2/(2b-a) 1/(b-1) - 1/(b+1) 1/(x-1) - 2/(x2-1)

Step-by-step solution

Rewrite the denominator of the second term.

Combine the fractions with the common denominator.

Factor the numerator using the difference of squares formula ( $x^2-y^2 = (x-y)(x+y)$ ).

Cancel out the common factor $(a-2b)$ .

Find a common denominator, which is $(b-1)(b+1)$ .

Perform the subtraction.

Simplify the numerator and the denominator.

Factor the denominator of the second term using the difference of squares formula.

Find a common denominator, which is $(x-1)(x+1)$ .

Perform the subtraction.

Simplify the numerator.

Cancel out the common factor $(x-1)$ .

Factor the numerators using the sum and difference of cubes formulas

Substitute the factored forms into the expression.

Cancel out common factors in each term.

Simplify the expression.

Find a common denominator, which is $(m+n)(m-n) = m^2-n^2$ .

Expand the numerators.

Substitute the expanded numerators back into the expression.

Combine the fractions.

Simplify the numerator.

Still stuck on this one?

More Mathematics Questions

Simplify the following algebraic expressions: a2/(a-2b) + 4b2/(2b-a) 1/(b-1) - 1/(b+1) 1/(x-1) - 2/(x2-1)

Step-by-step solution

Rewrite the denominator of the second term.

Combine the fractions with the common denominator.

Factor the numerator using the difference of squares formula (x2−y2=(x−y)(x+y)x^2-y^2 = (x-y)(x+y)x2−y2=(x−y)(x+y)).

Cancel out the common factor (a−2b)(a-2b)(a−2b).

Find a common denominator, which is (b−1)(b+1)(b-1)(b+1)(b−1)(b+1).

Perform the subtraction.

Simplify the numerator and the denominator.

Factor the denominator of the second term using the difference of squares formula.

Find a common denominator, which is (x−1)(x+1)(x-1)(x+1)(x−1)(x+1).

Perform the subtraction.

Simplify the numerator.

Cancel out the common factor (x−1)(x-1)(x−1).

Factor the numerators using the sum and difference of cubes formulas

Substitute the factored forms into the expression.

Cancel out common factors in each term.

Simplify the expression.

Find a common denominator, which is (m+n)(m−n)=m2−n2(m+n)(m-n) = m^2-n^2(m+n)(m−n)=m2−n2.

Expand the numerators.

Substitute the expanded numerators back into the expression.

Combine the fractions.

Simplify the numerator.

Still stuck on this one?

More Mathematics Questions

Factor the numerator using the difference of squares formula ( $x^2-y^2 = (x-y)(x+y)$ ).

Cancel out the common factor $(a-2b)$ .

Find a common denominator, which is $(b-1)(b+1)$ .

Find a common denominator, which is $(x-1)(x+1)$ .

Cancel out the common factor $(x-1)$ .

Find a common denominator, which is $(m+n)(m-n) = m^2-n^2$ .