Rational Expressions

Question 1

a ratio of two polynomials. Think of it as a fraction but instead of whole numbers, its numerator and denominator are polynomials.

Answer 1

Rational Expression

Question 2

Formally, a rational expression R(x) is the ratio of two polynomials P(x) and Q(x), such that the value of the polynomial Q(x) is not equal to 0 (because division by 0 is undefined).

Answer 2

R(x) = P(x)∕Q(x) , where Q(x) ≠ 0

Question 3

Example:Which of the following is/are rational expressions?

a).x²+7x-8/x²+3x+12
b).x²+2x+1
c).1/x²+2x+1
d).x-sqrt(2x)/x²+2x+1

Answer 3

Solution:

The expression ina)is a rational expression since both its numerator and denominator are polynomials.

The expression inb)is also a rational expression because x2+ 2x + 1 can be expressed as:

Take note thata constant can be considered a polynomial. Thus,b)has a numerator and denominator that are both polynomials.

The expression inc)is also a rational expression since its numerator (which is 1) is a polynomial while its denominator is also a polynomial.

The expression ind)is not a rational expression since its numerator is not a polynomial. Recall thatif a variable is under the radical sign, then the expression is not a polynomial.

Question 4

Simplifying Rational Expressions

Answer 4

Just like fractions, we can also reduce rational expressions into their simplest form. A rational expression is said to be in its simplest form if and only if its numerator and denominator have no common factor except 1.

For instance, let us take a look at the following rational expression:

If we factor both the numerator and denominator, you will notice that there’s acommon factorbetween them. That common factor isx:

We can cancel out the common factor:

What’s left with us is ⅖. Both 2 and 5 are prime and have no common factor except 1. Therefore, the simplified form of the rational expression in this example is ⅖.

Question 5

Here are the steps to simplify a rational expression:

Answer 5

Factor the numerator and the denominator

.Look for the common factors between the numerator and the denominator.

Cancel out the common factors between the numerator and the denominator.

Question 6

Example 1:Simplify the following rational expression:

18x³/3x

Answer 6

Solution:

1.Factor the numerator and the denominator.

2.Look for the common factors between the numerator and the denominator.

3.Cancel out the common factors between the numerator and the denominator.

Thus, the simplified form of the rational expression is 6x².

Question 7

Example 2:Simplify the following rational expression:

x²-2x/3x

Answer 7

Solution:

1.Factor the numerator and the denominator

We can factor outx2– 2xasx(x – 2)by factoring using theGreatest Common Factor (GCF).

2.Look for the common factors between the numerator and the denominator

3.Cancel out the common factors between the numerator and the denominator

=x-2/3

Question 8

Example 3:Simplify the following rational expression:

y²-16/y²-8y+16

Answer 8

Solution:

1.Factor the numerator and the denominator

Since y2– 16 is adifference of the two squares, we can factor it as (y + 4)(y – 4). On the other hand, y2– 8y + 16 is aperfect square trinomialthat we can factor as (y – 4)(y – 4).

2.Look for the common factors between the numerator and the denominator

3.Cancel out the common factors between the numerator and the denominator

=y+4/y-4

Question 9

Example 4:Simplify the following rational expression:

x²-16x+64/2x-16

Answer 9

Solution:

1.Factor the numerator and the denominator

Since x2– 16x + 64 is a perfect square trinomial, we can factor it as (x – 8)(x – 8). Meanwhile, we can factor 2x – 16 as 2(x – 8) using its GCF.

2.Look for the common factors between the numerator and the denominator

3.Cancel out the common factors between the numerator and the denominator

=x-8/2

Question 10

Example 5:Simplify the following rational expression:

a²+7a+10/a+5

Answer 10

Solution:

1.Factor the numerator and the denominator

We can factor a2+ 7a + 10 as (a + 5)(a + 2).

2.Look for the common factors between the numerator and the denominator

3.Cancel out the common factors between the numerator and the denominator

Therefore, the simplified form of the rational expression isa + 2.

Question 11

Example 6:Simplify the following rational expression:
(n²-16)(n+2)/n²+6n+8

Answer 11

Solution:

1.Factor the numerator and the denominator

We can factor n2– 16 as (n + 4)(n – 4). Meanwhile, we can factor n2+ 6n + 8 as (n + 4)(n + 2).

2.Look for the common factors between the numerator and the denominator

3.Cancel out the common factors between the numerator and the denominator

Thus, the answer to this example isn – 4.

Question 12

How To Find the Least Common Denominator (LCD) of Rational Expressions?

Answer 12

To find the LCD of rational expressions, follow these steps:

Factor the denominators of the rational expressions.

Write the factors of the denominators. Match the common factors in columns.

Bring down each factor in every column. Common factors in the column must be brought down also.

Multiply the factors you brought down. The resulting expression is the LCD.

Question 13

Example 1: determine the LCD of 2x/x-1 and x/x²-1

Answer 13

Solution:

The denominators of the given expressions arex – 1andx2– 1. Our task is to determine their Least Common Denominator using the steps above:

1.Factor the denominators of the rational expressions.

x – 1cannot be factored further. Meanwhile, sincex2– 1is adifference of two squares, we can factor it as(x + 1)(x – 1).

2.Write the factors of the denominators. Match the common factors in columns

3.Bring down each factor in every column. Common factors in the column must be brought down also

4.Multiply the factors you brought down. The resulting expression is the LCD

Thus, the LCD is(x + 1)(x – 1)orx² – 1.

Question 14

Example 2: Determine the LCD of 5x-2/x²+7x+10 and
x/x²+4x+4

Answer 14

Solution:

1.Factor the denominators of the rational expressions.

x2+ 7x + 10can be factored as(x + 5)(x + 2).Meanwhile,x2+ 4x + 4can be factored as(x + 2)(x + 2).

2.Write the factors of the denominators. Match the common factors in columns

3.Bring down each factor in every column. Common factors in the column must be brought down also

4.Multiply the factors you brought down. The resulting expression is the LCD

Based on our computations above, the LCD ofx2+ 7x + 10andx2+ 4x + 4is(x + 2)(x + 2)(x + 5).

Note: When we are determining the LCD of two rational expressions, it is advisable to write the obtained LCD in factored form since expressions are much easier to multiply and divide if they are in factored form.

Make sure that you already mastered the skill of determining the LCD of rational expressions before proceeding to the actual process of adding and subtracting them.

Question 15

Here are the steps in adding rational expressions with the same denominator:

Answer 15

Add the numerators of the rational expressions. The resulting expression is the numerator of the answer.

Copy the common denominator and use it as the denominator of your answer.

Simplify the resulting rational expression, if possible.

Question 16

Example 1: add 4x+2/x+1 and 3x/x+1

Answer 16

Solution:

1.Add the numerators of the rational expressions. The resultingexpression is the numerator of the answer

2.Copy the common denominator and use it as the denominator of your answer

3.Simplify the resulting rational expression, if possible

In this case, we can’t simplify the resulting rational expression further so it automatically becomes the final answer.

7x+2/x+1

Question 17

Example 2: add 2x-4/x-5 and x²+x-1/x-5

Answer 17

Solution:

Example 3:

Solution:

=5x+1/x-2

Question 18

Example 4: calculate m²-2m/m²2m+1 + 2m²/m²+2m+1

Answer 18

Solution:
=3m²-2m/m²+2m+1

Question 19

Example 5:

Compute k²+3/k-1 + k²-2/k-1

Answer 19

Solution:

2k²+1/k-1

Question 20

To add or subtract rational expressions with different denominators, follow these steps:

Answer 20

Determine the LCD of the rational expressions.

Express the given rational expressions using the LCD you have obtained by dividing the LCD by the denominator of the rational expression and then multiplying the result to the numerator of the rational expression. The results will be the new numerators of the rational expressions.

Add or subtract the rational expressions you have obtained from the second step. Simplify the resulting expression, if possible.

Question 21

Example 1: add 3/2x and 5/3x²

Answer 21

Solution:

Using the steps we have mentioned above on adding and subtracting rational expressions with different denominators:

1.Determine the LCD of the rational expressions.

2.Express the given rational expressions using the LCD you have obtained by dividing the LCD by the denominator of the rational expression and then multiplying the result to the numerator of the rational expression. The results will be the new numerators of rational expressions.

3.Add or subtract the rational expressions you have obtained from the second step. Simplify the resulting expression, if possible.

=9x+10/6x²

Question 22

Example 2: Compute for the sum of x+1/x-1 and 2/x+1

Answer 22

Solution:

1.Determine the LCD of the rational expressions.

The LCD is (x – 1)(x + 1).

2.Express the given rational expressions using the LCD you have obtained by dividing the LCD by the denominator of the rational expression and then multiplying the result to the numerator of the rational expression. The results will be the new numerators of rational expressions.

3. Add or subtract the rational expressions you have obtained from the second step. Simplify the resulting expression, if possible.

=x²+3/(x-1)(x+1)

Question 23

Example 3:

Determine the difference of x-1/2x+8 and x+2/x+4

Answer 23

Solution:

1.Determine the LCD of the rational expressions

2.Express the given rational expressions using the LCD you have obtained by dividing the LCD by the denominator of the rational expression and then multiplying the result to the numerator of the rational expression. The results will be the new numerators of the rational expressions.

3.Add or subtract the rational expressions you have obtained from the second step. Simplify the resulting expression, if possible.

-x-5/2x+8

Question 24

The steps in multiplying rational expressions are actually similar to the steps inmultiplying fractions. Here are the steps:

Answer 24

Multiply the numerators of the rational expressions. Write the answer as the numerator of the resulting expression.

Multiply the denominators of the rational expressions. Write the answer as the denominator of the resulting expression.

Simplify the resulting expression, if possible.

Question 25

Example: multiply 5/x and x²/x-9

Answer 25

Solution:

Using the steps in multiplying rational expressions:

1.Multiply the numerators of the rational expressions. Write the answer as the numerator of the resulting expression.

2.Multiply the denominators of the rational expressions. Write the answer as the denominator of the resulting expression.

We will not performdistributive propertyin this case since we are simplifying the expression in the next step.

3.Simplify the resulting expression, if possible.

=5x/x-9

Question 26

Example 1:Apply the cancellation method to calculate the product of

x²-4x+4/9x • 3x/2x-4

Answer 26

Solution:

=x-2/6

Question 27

Example 2:

Multiply a²-6ab+9b²/5y-15 • 2y-6/a-3b

Answer 27

Solution:

=2a - 6b / 5

Question 28

If you still rememberhow to divide fractions, then dividing rational expressions will be easier because the steps are actually similar. Otherwise, here are the steps you need to remember when dividing rational expressions:

Answer 28

Get the reciprocal of the divisor or the second rational expression.

Multiply the rational expression you have obtained in Step 1 to the first rational expression.

Simplify the result, if possible.

Question 29

Example 1: Divide x+8/x by 3x/x-8

Answer 29

Solution:

1.Get the reciprocal of the divisor or the second rational expression.

2.Multiply the rational expression you have obtained in Step 1 to the first rational expression.

3. Simplify the result, if possible.

=x²-64/3x²

Question 30

Example 2: Divide 2x-18/x+4 by x+4/x

Answer 30

Solution:

=2x²-18x/x²+8x+16

Question 31

1) Simplify:𝑥2− 1/𝑥2+ 2𝑥 + 1

a)𝑥 + 1/𝑥 − 1
b)𝑥 − 1/𝑥 + 1
c)𝑥/𝑥 + 1
d)𝑥 + 1/𝑥

Answer 31

B

Question 32

2) Add: 3/x and 5/x-1

a)2𝑥 + 3/𝑥2 + 1
b)8𝑥 − 3/𝑥
c)8𝑥 − 3/𝑥2 − 𝑥
d)𝑥2 − 𝑥/2𝑥 − 3

Answer 32

C

Question 33

3) Multiply and 𝑥 + 1/𝑥 and 3/𝑥 + 1

a) 3/x
b) 6/x2
c) x/3
d) x2/6

Answer 33

A

Question 34

4) Add: 2 + 𝑥/2𝑥2 + 3𝑥 − 1 + 𝑥 − 1/2𝑥2 + 3𝑥 − 1

a)2𝑥 + 1/2𝑥2 + 3𝑥 − 1
b)1 − 2𝑥/2𝑥2 + 3𝑥 − 1
c)2𝑥 − 1/2𝑥2 + 3𝑥 − 1
d)𝑥 − 2/2𝑥2 + 3𝑥 − 1

Answer 34

A

Question 35

5) Divide by 3/y by 27/y²

a) 9/y
b) y2/9
c) 9/y2
d) y/9

Answer 35

D