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Add & Subtract Rational Expressions with Unlike Denominators>
Add & Subtract Rational Expressions with Unlike Denominators 2Add & Subtract Rational Expressions with Unlike Denominators 2
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Question 1 of 4
1. Question
Perform the operation`4/(3n+6)+n/(n^2-4)`Hint
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Adding and Subtracting Rational Expressions
`a/c +- b/c = (a+-b)/c`First, expand the denominators.`4/(3n+6)+n/(n^2-4)` `=` `4/(3(n+2))+n/((n+2)(n-2))` Next, identify the LCD.Since we have different denominators, the LCD would be `(3)(n+2)(n-2)`.Convert into same denominators using the LCD.$$\frac {4}{3n+6}+ \frac {n}{n^2-4}$$ $$=$$ $$\frac {4}{3(n+2)}+ \frac {n}{(n-2)(n+2)}$$ $$=$$ $$\frac {4 \times \color{red}{(n-2)}}{3(n+2) \times \color{red}{(n-2)}}+ \frac {n \times \color{red}{3}}{(n-2)(n+2) \times \color {red}{3}}$$ LCD is `3(n-2)(n+2)` `=` `(4(n-2))/(3(n-2)(n+2))+ (3n)/(3(n-2)(n+2))` Simplify `=` `(4(n-2)+3n)/(3(n-2)(n+2))` Combine numerators `=` `(4n-8+3n)/(3(n-2)(n+2))` Distribute `4` inside parenthesis `=` `(7n-8)/(3(n-2)(n+2))` Combine like terms `(7n-8)/(3(n-2)(n+2))` -
Question 2 of 4
2. Question
Perform the operation`5/(4x-8)-(3x)/(x+2)`Hint
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Adding and Subtracting Rational Expressions
`a/c +- b/c = (a+-b)/c`First, identify the LCD.Since we have different denominators, the LCD would be `(4x-8)(x+2)`.Convert into same denominators using the LCD.$$\frac {5}{4x-8}- \frac {3x}{x+2}$$ $$=$$ $$\frac {5 \times \color{red}{(x+2)}}{(4x-8) \times \color {red}{(x+2)}}- \frac {3x \times \color{red}{(4x-8)}}{(x+2) \times \color{red}{(4x-8)}}$$ LCD is `(4x-8)(x+2)` `=` `(5(x+2)-3x(4x-8))/((4x-8)(x+2))` Combine the numerators `=` `(5x+10-12x^2+24x)/((4x-8)(x+2))` Distribute the terms inside parenthesis `=` `(-12x^2+29x+10)/((4x-8)(x+2))` Combine like terms `(-12x^2+29x+10)/((4x-8)(x+2))` -
Question 3 of 4
3. Question
Perform the operation`(3m+1)/(m-1)^2-(m-2)/(m^2+4m-5)`Hint
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Adding and Subtracting Rational Expressions
`a/c +- b/c = (a+-b)/c`First, expand the denominators starting with the first term.`(3m+1)/(m-1)^2-(m-2)/(m^2+4m-5)` `=` `(3m+1)/((m-1)(m-1))-(m-2)/(m^2+4m-5)` Since the denominator of the second term is in standard form `(``a``x^2+``b``x+``c``=0)` we can factorise using the cross method.`m^2+``4``m``-5``=0`To factorise, we need to find two numbers that add to `4` and multiply to `-5``5` and `-1` fit both conditions`5 – 1` `=` `4` `5 xx -1` `=` `-5` 
Read across to get the factors.`(m+5)(m-1)=0`Copy the factors of `m^2+4m-5`.`(3m+1)/(m-1)^2-(m-2)/(m^2+4m-5)` `=` `(3m+1)/((m-1)(m-1))-(m-2)/((m+5)(m-1))` Next, identify the LCD.`m-1` is common in the denominators, so the LCD would be `(m-1)(m-1)(m+5)`.Convert into same denominators using the LCD.$$=$$ $$\frac {(3m+1) \times \color{red}{(m+5)}}{(m-1)(m-1) \times \color{red}{(m+5)}}+ \frac {(m-2) \times \color{red}{(m-1)}}{(m+5)(m-1) \times \color{red}{(m-1)}}$$ LCD is `(m-1)(m-1)(m+5)` `=` `((3m+1)(m+5)+(m-2)(m-1))/((m-1)(m-1)(m+5))` Combine the numerators `=` `((3m^2+16m+5)+(m^2-3m+2))/((m-1)(m-1)(m+5))` Distribute the terms `=` `(4m^2+13m+7)/((m-1)(m-1)(m+5))` Combine like terms `(4m^2+13m+7)/((m-1)(m-1)(m+5))` -
Question 4 of 4
4. Question
Perform the operation`(a-3)/(a^2+6a+9)-(a+3)/(a-3)`Hint
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Exceptional!
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Adding and Subtracting Rational Expressions
`a/c +- b/c = (a+-b)/c`Since the denominator of the first term is in standard form `(``a``x^2+``b``x+``c``=0)` we can factorise using the cross method.`a^2+``6``a+``9``=0`To factorise, we need to find two numbers that add to `6` and multiply to `9``3` and `-3` fit both conditions`3 + 3` `=` `6` `3 xx 3` `=` `9` 
Read across to get the factors.`(a+3)(a+3)=0`Copy the factors to the original expression.`(a-3)/(a^2+6a+9)-(a+3)/(a-3)` `=` `(a-3)/((a+3)(a+3))-(a+3)/(a-3)` Next, identify the LCD.There is no common term in the denominators, so the LCD would be `(a+3)(a+3)(a-3)`.Convert into same denominators using the LCD.$$\frac {a-3}{a^2+6a+9}- \frac {a+3}{a-3}$$ $$=$$ $$\frac {(a-3) \times \color{red}{(a-3)}}{(a+3)(a+3) \times \color {red}{(a-3)}}- \frac {(a+3) \times \color{red}{(a+3)(a+3)}}{(a-3) \times \color{red}{(a+3)(a+3)}}$$ Multiply to come up with same denominators `=` `((a-3)(a-3)-(a+3)(a+3)(a+3))/((a+3)(a+3)(a-3))` `=` `((a^2-6a+9)-(a^3+9a^2+27a+27))/((a+3)(a+3)(a-3))` Distribute the factors `=` `(a^2-6a+9-a^3-9a^2-27a-27)/((a+3)(a+3)(a-3))` Distribute the negative sign `=` `(-a^3-8a^2-33a-18)/((a+3)(a+3)(a-3))` Combine like terms `(-a^3-8a^2-33a-18)/((a+3)(a+3)(a-3))`