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Adding rational expression: unlike denominators

Sal rewrites (5x)/(2x-3)+(-4x²)/(3x+1) as (-8x³+27x²+5x)/(2x-3)(3x+1).

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• What if we did not distribute the numbers when adding them and leave them as factors. and then we can cancel out the numbers.
• You CAN leave sums of products as factors if they share a common factor. So 3(x-4) + 5(x-4) can be added to get 8(x-4). But you cannot add two different products of two different sums without distributing them first. .For example 3(x+4) + 5(2x-5) share no common factors. You cannot add them without distributing first and then combining like terms. You have to distribute to get 3x +12 +10x -25 which then simplifies to 13x-13 and could be re factored to 13(x-1). You cannot see those two factors of the original expression, 13 and (x-1), until you do the intervening arithmetic.
• Why wouldn't you multiply the denominators together at the end?
• Often you need to factor the equations to see if there are any things you can cancel out to simplify it. I don't know about you but my teacher tells me to leave it like that in case there are further calculations.
• none of this makes sense.
• They all give numbers, you're just multiplying the expressions while keeping them the same, so they will give numbers back
• Why didn't he factor by grouping to see if something cancels out?
• Why would you not factor out the x in the numerator and leave it like x(-8x²+27x+5) over (2x-3)(3x+1)?
• Can't you just combine the like terms on the bottom to get 5x - 3 and then use that as a common denominator?
• Sorry, but no.
Each of the denominators represents a single quantity [ (2x-3) and (3x+1) ]---it's the same as if we were adding 4/7 + 3/5. Their common denominator wouldn't be (7 + 5), it would have to have FACTORS of 7 and 5 --- in other words 7 * 5.
That's why the common denominator turns out to be (2x-3)(3x+1).
• in x/2 - x/4 = 6 why do we multiply both sides by 4?
• Equations must be kept in balance. Visualize a scale that's in balance. If both sides have 2 pounds on them and you only multiply one side by 4, that side now has 8 pounds and the other still has 2. So, the scale is not in balance.

Unless you are simplifying one side (just reorganizing, not changing the value), then you must do the operation to both sides of the equation.

Hope this helps.
• I'm practicing SAT Test on Khan Academy and I do not get why I keep getting different answers. The question of the problem is 7/x-1 + 6/x^2-12x+11. To the answer of the problem, it shows that you can factor x^2-12x+11 denominator but when I multiply the denominators x-1 and x^2-12x+11 I would get different answer.
(7/x-1)x^2-12x+11 + (6/x^2-12x+11)x-1. The answer I keep getting is
7x^2-78x+76/(x-1)(x^2-12x+11). And the actual answer of the problem is
7x-71/(x-1)(x-11). Don't get why I'm getting this answer. Thank you.
• To find a common denominator, you need to first factor denominators where possible. You didn't factor the x^2-12x+11 so you aren't finding the smallest common denominator. The factors of x^2-12x+11 are (x-1)(x-11). So, the LCD = (x-1)(x-11) rather than your version of (x-1)(x^2-12x+11).

Since you didn't use the smallest LCD, your answer is coming out as a fraction that is not fully reduced. You need to factor your numerator and the denominator to fully reduce the fraction.

You also have an error in your numerator
7(x^2-12x+11) = 7x^2-84x+77
6(x-1) = 6x-6
7x^2-84x+77+6x-6 = 7x^2-84x+6x+77-6 = 7x^2-78x+71
You have 76 rather than 71 as your last terms.

Hope this helps.