Adding and Subtracting Rational Expressions

0:00How do we add 3 over x and 2 over x plus 1?

0:05Each is a rational expression: a fraction with a polynomial on top and on the bottom.

0:11Add them just like fractions. To add one half and one third, we need a common denominator: one bottom number for both.

0:202 and 3 both go into 6, so one half is 3 sixths, one third is 2 sixths, and the sum is 5 sixths.

0:30Here the denominators are x and x plus 1, so a common denominator is x times the bracket x plus 1.

0:39Multiply the first fraction, top and bottom, by x plus 1, and the second by x. Their values stay the same.

0:48Now add the numerators, the top parts, and keep the common denominator.

0:533 times the bracket x plus 1 is 3x plus 3. Add 2x to get 5x plus 3.

1:02A denominator can never be 0, so x cannot be 0 or negative 1.

1:09Check with x equals 1. The question gives 3 plus 1, which is 4. The answer gives 8 over 2, which is also 4.

1:19Subtracting works the same way: 5 over x minus 2, take away x plus 1 over x minus 2.

1:27The denominators already match, so subtract the whole second numerator, in brackets.

1:34The minus sign applies to both terms in the brackets: 5 minus x minus 1.

1:415 minus 1 is 4, so the answer is 4 minus x over x minus 2, and x cannot be 2.

1:50A common mistake is to subtract only the x, giving 6 minus x. At x equals 3 that gives 3, but the true answer is 1.

2:01To add or subtract rational expressions, use a common denominator, then combine the whole numerators.