Simplify the following algebraic expressions: (1)/(x-y) − (1)/(x+y) + (2x)/(x2-y2) and (x+2)/(1+x+x2)…

Question

Simplify the following algebraic expressions: (1)/(x-y) − (1)/(x+y) + (2x)/(x2-y2) and (x+2)/(1+x+x2) − (x−2)/(1+x+x2) − (2x2)/(1+x2+x4)

ScanSolve AI Tutor · Accepted Answer

here are the solutions to the problems:

1)

\frac{1}{x-y} - \frac{1}{x+y} + \frac{2x}{x^2-y^2}

Find a common denominator for the first two terms, which is $(x-y)(x+y) = x^2-y^2$ .
$\frac{1(x+y)}{(x-y)(x+y)} - \frac{1(x-y)}{(x+y)(x-y)} + \frac{2x}{x^2-y^2}$
Combine the first two fractions.
$\frac{(x+y) - (x-y)}{x^2-y^2} + \frac{2x}{x^2-y^2}$
Simplify the numerator of the first combined fraction.
$\frac{x+y-x+y}{x^2-y^2} + \frac{2x}{x^2-y^2} = \frac{2y}{x^2-y^2} + \frac{2x}{x^2-y^2}$
Combine all fractions since they now have a common denominator.
$\frac{2y+2x}{x^2-y^2}$
Factor out 2 from the numerator and factor the denominator using the difference of squares formula ( $x^2-y^2 = (x-y)(x+y)$ ).
$\frac{2(x+y)}{(x-y)(x+y)}$
Cancel out the common factor $(x+y)$ .
$\frac{2}{x-y}$
The simplified expression is $\boxed{\frac{2}{x-y}}$ .

2)
$\frac{x+2}{1+x+x^2} - \frac{x-2}{1+x+x^2} - \frac{2x^2}{1+x^2+x^4}$
Combine the first two terms as they share a common denominator.
$\frac{(x+2) - (x-2)}{1+x+x^2} - \frac{2x^2}{1+x^2+x^4}$
Simplify the numerator of the first term.
$\frac{x+2-x+2}{1+x+x^2} - \frac{2x^2}{1+x^2+x^4} = \frac{4}{1+x+x^2} - \frac{2x^2}{1+x^2+x^4}$
Factor the denominator of the second term using the identity $a^4+a^2b^2+b^4 = (a^2+ab+b^2)(a^2-ab+b^2)$ .
Here, $a=x$ and $b=1$ .
$1+x^2+x^4 = (x^2+x+1)(x^2-x+1)$
Substitute this into the expression:
$\frac{4}{1+x+x^2} - \frac{2x^2}{(1+x+x^2)(1-x+x^2)}$
Find a common denominator, which is $(1+x+x^2)(1-x+x^2)$ .
$\frac{4(1-x+x^2)}{(1+x+x^2)(1-x+x^2)} - \frac{2x^2}{(1+x+x^2)(1-x+x^2)}$
Combine the fractions.
$\frac{4(1-x+x^2) - 2x^2}{(1+x+x^2)(1-x+x^2)}$
Expand and simplify the numerator.
$\frac{4-4x+4x^2 - 2x^2}{(1+x+x^2)(1-x+x^2)} = \frac{4-4x+2x^2}{1+x^2+x^4}$
Factor out 2 from the numerator.
$\frac{2(x^2-2x+2)}{x^4+x^2+1}$
The simplified expression is $\boxed{\frac{2(x^2-2x+2)}{x^4+x^2+1}}$ .

3)
$\frac{1}{b+c} + \frac{b-c}{b^2-c^2}$
Factor the denominator of the second term using the difference of squares formula ( $b^2-c^2 = (b-c)(b+c)$ ).
$\frac{1}{b+c} + \frac{b-c}{(b-c)(b+c)}$
Cancel out the common factor $(b-c)$ in the second term.
$\frac{1}{b+c} + \frac{1}{b+c}$
Combine the fractions as they have a common denominator.
$\frac{1+1}{b+c} = \frac{2}{b+c}$
The simplified expression is $\boxed{\frac{2}{b+c}}$ .

Simplify the following algebraic expressions: (1)/(x-y) - (1)/(x+y) + (2x)/(x2-y2) and (x+2)/(1+x+x2) - (x-2)/(1+x+x2) - (2x2)/(1+x2+x4)

Step-by-step solution

Find a common denominator for the first two terms, which is $(x-y)(x+y) = x^2-y^2$ .

Combine the first two fractions.

Simplify the numerator of the first combined fraction.

Combine all fractions since they now have a common denominator.

Factor out 2 from the numerator and factor the denominator using the difference of squares formula ( $x^2-y^2 = (x-y)(x+y)$ ).

Cancel out the common factor $(x+y)$ .

Combine the first two terms as they share a common denominator.

Simplify the numerator of the first term.

Factor the denominator of the second term using the identity $a^4+a^2b^2+b^4 = (a^2+ab+b^2)(a^2-ab+b^2)$ .

Find a common denominator, which is $(1+x+x^2)(1-x+x^2)$ .

Combine the fractions.

Expand and simplify the numerator.

Factor out 2 from the numerator.

Factor the denominator of the second term using the difference of squares formula ( $b^2-c^2 = (b-c)(b+c)$ ).

Cancel out the common factor $(b-c)$ in the second term.

Combine the fractions as they have a common denominator.

Still stuck on this one?

More Mathematics Questions

Simplify the following algebraic expressions: (1)/(x-y) - (1)/(x+y) + (2x)/(x2-y2) and (x+2)/(1+x+x2) - (x-2)/(1+x+x2) - (2x2)/(1+x2+x4)

Step-by-step solution

Find a common denominator for the first two terms, which is (x−y)(x+y)=x2−y2(x-y)(x+y) = x^2-y^2(x−y)(x+y)=x2−y2.

Combine the first two fractions.

Simplify the numerator of the first combined fraction.

Combine all fractions since they now have a common denominator.

Factor out 2 from the numerator and factor the denominator 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 (x+y)(x+y)(x+y).

Combine the first two terms as they share a common denominator.

Simplify the numerator of the first term.

Factor the denominator of the second term using the identity a4+a2b2+b4=(a2+ab+b2)(a2−ab+b2)a^4+a^2b^2+b^4 = (a^2+ab+b^2)(a^2-ab+b^2)a4+a2b2+b4=(a2+ab+b2)(a2−ab+b2).

Find a common denominator, which is (1+x+x2)(1−x+x2)(1+x+x^2)(1-x+x^2)(1+x+x2)(1−x+x2).

Combine the fractions.

Expand and simplify the numerator.

Factor out 2 from the numerator.

Factor the denominator of the second term using the difference of squares formula (b2−c2=(b−c)(b+c)b^2-c^2 = (b-c)(b+c)b2−c2=(b−c)(b+c)).

Cancel out the common factor (b−c)(b-c)(b−c) in the second term.

Combine the fractions as they have a common denominator.

Still stuck on this one?

More Mathematics Questions

Find a common denominator for the first two terms, which is $(x-y)(x+y) = x^2-y^2$ .

Factor out 2 from the numerator and factor the denominator using the difference of squares formula ( $x^2-y^2 = (x-y)(x+y)$ ).

Cancel out the common factor $(x+y)$ .

Factor the denominator of the second term using the identity $a^4+a^2b^2+b^4 = (a^2+ab+b^2)(a^2-ab+b^2)$ .

Find a common denominator, which is $(1+x+x^2)(1-x+x^2)$ .

Factor the denominator of the second term using the difference of squares formula ( $b^2-c^2 = (b-c)(b+c)$ ).

Cancel out the common factor $(b-c)$ in the second term.