Learning Objectives
By the end of this section, you will be able to:
Add and subtract rational expressions with a common denominator
Add and subtract rational expressions whose denominators are opposites
Find the least common denominator of rational expressions
Add and subtract rational expressions with unlike denominators
Add and subtract rational functions
Be Prepared 7.4
Before you get started, take this readiness quiz.
Add: 710+815.710+815. If you missed this problem, review Example 1.29.
Be Prepared 7.5
Subtract: 3x4−89.3x4−89. If you missed this problem, review Example 1.28.
Be Prepared 7.6
Subtract: 6(2x+1)−4(x−5).6(2x+1)−4(x−5). If you missed this problem, review Example 1.56.
Add and Subtract Rational Expressions with a Common Denominator
What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.
It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.
Rational Expression Addition and Subtraction
If p, q, and r are polynomials where r≠0,r≠0, then
To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result over the common denominator.
We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.
Remember, too, we do not allow values that would make the denominator zero. What value of x should be excluded in the next example?
Example 7.13
Add: 11x+28x+4+x2x+4.11x+28x+4+x2x+4.
Solution
Since the denominator is x+4,x+4, we must exclude the value x=−4.x=−4.
row: 11x+28x+4+x2x+4,x≠−411x+28x+4+x2x+4,x≠−4
row: The fractions have a common denominator,so add the numerators and place the sumover the common denominator. | 11x+28+x2x+411x+28+x2x+4
row: Write the degrees in descending order. | x2+11x+28x+4x2+11x+28x+4
row: Factor the numerator. | (x+4)(x+7)x+4(x+4)(x+7)x+4
row: Simplify by removing common factors. | (x+4)(x+7)x+4(x+4)(x+7)x+4
row: Simplify. | x+7x+7
The expression simplifies to x+7x+7 but the original expression had a denominator of x+4x+4 so x≠−4.x≠−4.
Try It 7.25
Simplify: 9x+14x+7+x2x+7.9x+14x+7+x2x+7.
Try It 7.26
Simplify: x2+8xx+5+15x+5.x2+8xx+5+15x+5.
To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator. Be careful of the signs when you subtract a binomial or trinomial.
Example 7.14
Subtract: 5x2−7x+3x2−3x-18−4x2+x−9x2−3x-18.5x2−7x+3x2−3x-18−4x2+x−9x2−3x-18.
Solution
row: 5x2−7x+3x2−3x-18−4x2+x−9x2−3x-185x2−7x+3x2−3x-18−4x2+x−9x2−3x-18
row: Subtract the numerators and place thedifference over the common denominator. | 5x2−7x+3−(4x2+x−9)x2−3x-185x2−7x+3−(4x2+x−9)x2−3x-18
row: Distribute the sign in the numerator. | 5x2−7x+3−4x2−x+9x2−3x−185x2−7x+3−4x2−x+9x2−3x−18
row: Combine like terms. | x2−8x+12x2−3x−18x2−8x+12x2−3x−18
row: Factor the numerator and the denominator. | (x−2)(x−6)(x+3)(x−6)(x−2)(x−6)(x+3)(x−6)
row: Simplify by removing common factors. | (x−2)(x−6)(x+3)(x−6)(x−2)(x−6)(x+3)(x−6)
row: (x−2)(x+3)(x−2)(x+3)
Try It 7.27
Subtract: 4x2−11x+8x2−3x+2−3x2+x−3x2−3x+2.4x2−11x+8x2−3x+2−3x2+x−3x2−3x+2.
Try It 7.28
Subtract: 6x2−x+20x2−81−5x2+11x−7x2−81.6x2−x+20x2−81−5x2+11x−7x2−81.
Add and Subtract Rational Expressions Whose Denominators are Opposites
When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by −1−1.−1−1.
Let’s see how this works.
row: Multiply the second fraction by −1−1.−1−1.
row: The denominators are the same.
row: Simplify.
Be careful with the signs as you work with the opposites when the fractions are being subtracted.
Example 7.15
Subtract: m2−6mm2−1−3m+21−m2.m2−6mm2−1−3m+21−m2.
Solution
row: The denominators are opposites, so multiply thesecond fraction by −1−1.−1−1.
row: Simplify the second fraction.
row: The denominators are the same. Subtract the numerators.
row: Distribute.
row: Combine like terms.
row: Factor the numerator and denominator.
row: Simplify by removing common factors.
row: Simplify.
Try It 7.29
Subtract: y2−5yy2−4−6y−64−y2.y2−5yy2−4−6y−64−y2.
Try It 7.30
Subtract: 2n2+8n−1n2−1−n2−7n−11−n2.2n2+8n−1n2−1−n2−7n−11−n2.
Find the Least Common Denominator of Rational Expressions
When we add or subtract rational expressions with unlike denominators, we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.
Let’s look at this example: 712+518.712+518. Since the denominators are not the same, the first step was to find the least common denominator (LCD).
To find the LCD of the fractions, we factored 12 and 18 into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.
When we add numerical fractions, once we found the LCD, we rewrote each fraction as an equivalent fraction with the LCD by multiplying the numerator and denominator by the same number. We are now ready to add.
We do the same thing for rational expressions. However, we leave the LCD in factored form.
How To
Find the least common denominator of rational expressions.
Step 1. Factor each denominator completely.
Step 2. List the factors of each denominator. Match factors vertically when possible.
Step 3. Bring down the columns by including all factors, but do not include common factors twice.
Step 4. Write the LCD as the product of the factors.
Remember, we always exclude values that would make the denominator zero. What values of xx should we exclude in this next example?
Example 7.16
ⓐ Find the LCD for the expressions 8x2−2x−3,3xx2+4x+38x2−2x−3,3xx2+4x+3 and ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.
Solution
ⓐ
row: Find the LCD for 8x2−2x−3,3xx2+4x+3.8x2−2x−3,3xx2+4x+3.
row: Factor each denominator completely, lining up common factors.Bring down the columns.
row: Write the LCD as the product of the factors.
ⓑ
row: Factor each denominator.
ਹਰ ਇੱਕ ਹਰ (denominator) ਨੂੰ 'ਗੁੰਮ' ਹੋਏ LCD ਗੁਣਕ ਨਾਲ ਗੁਣਾ ਕਰੋ ਅਤੇ ਹਰ ਇੱਕ ਅੰਸ਼ (numerator) ਨੂੰ ਉਸੇ ਗੁਣਕ ਨਾਲ ਗੁਣਾ ਕਰੋ।
ਅੰਸ਼ਾਂ ਨੂੰ ਸਰਲ ਕਰੋ।
ਪ੍ਰਯੋਗ ਕਰੋ 7.31
ⓐ ਪਦਾਂ 2x²−x−12, 1x²−16 ਲਈ LCD ਲੱਭੋ। ⓑ ਉਹਨਾਂ ਨੂੰ ਸਭ ਤੋਂ ਛੋਟੇ ਸਾਂਝੇ ਹਰ ਨਾਲ ਬਰਾਬਰ ਦੇ ਪਰਿਮੇਯ ਪਦਾਂ ਵਜੋਂ ਮੁੜ ਲਿਖੋ।
ਪ੍ਰਯੋਗ ਕਰੋ 7.32
ⓐ ਪਦਾਂ 3xx²−3x–10, 5x²+3x+2 ਲਈ LCD ਲੱਭੋ। ⓑ ਉਹਨਾਂ ਨੂੰ ਸਭ ਤੋਂ ਛੋਟੇ ਸਾਂਝੇ ਹਰ ਨਾਲ ਬਰਾਬਰ ਦੇ ਪਰਿਮੇਯ ਪਦਾਂ ਵਜੋਂ ਮੁੜ ਲਿਖੋ।
ਵੱਖਰੇ ਹਰਾਂ ਵਾਲੇ ਪਰਿਮੇਯ ਪਦਾਂ ਨੂੰ ਜੋੜੋ ਅਤੇ ਘਟਾਓ।
ਹੁਣ ਸਾਡੇ ਕੋਲ ਵੱਖਰੇ ਹਰਾਂ ਵਾਲੇ ਪਰਿਮੇਯ ਪਦਾਂ ਨੂੰ ਜੋੜਨ ਜਾਂ ਘਟਾਉਣ ਲਈ ਸਾਰੇ ਲੋੜੀਂਦੇ ਕਦਮ ਹਨ।
ਉਦਾਹਰਨ 7.17
ਵੱਖਰੇ ਹਰਾਂ ਵਾਲੇ ਪਰਿਮੇਯ ਪਦਾਂ ਨੂੰ ਕਿਵੇਂ ਜੋੜਨਾ ਹੈ।
ਜੋੜੋ: 3x−3+2x−2।
ਹੱਲ
ਪ੍ਰਯੋਗ ਕਰੋ 7.33
ਜੋੜੋ: 2x−2+5x+3।
ਪ੍ਰਯੋਗ ਕਰੋ 7.34
ਜੋੜੋ: 4m+3+3m+4।
ਪਰਿਮੇਯ ਪਦਾਂ ਨੂੰ ਜੋੜਨ ਲਈ ਵਰਤੇ ਗਏ ਕਦਮ ਇੱਥੇ ਸੰਖੇਪ ਕੀਤੇ ਗਏ ਹਨ।
ਕਿਵੇਂ ਕਰਨਾ ਹੈ
ਪਰਿਮੇਯ ਪਦਾਂ ਨੂੰ ਜੋੜੋ ਜਾਂ ਘਟਾਓ।
ਕਦਮ 1. ਨਿਰਧਾਰਤ ਕਰੋ ਕਿ ਕੀ ਪਦਾਂ ਦਾ ਸਾਂਝਾ ਹਰ ਹੈ। ਹਾਂ – ਕਦਮ 2 'ਤੇ ਜਾਓ। ਨਹੀਂ – ਹਰ ਪਰਿਮੇਯ ਪਦ ਨੂੰ LCD ਨਾਲ ਬਰਾਬਰ ਦੇ ਪਰਿਮੇਯ ਪਦ ਵਜੋਂ ਮੁੜ ਲਿਖੋ। LCD ਲੱਭੋ। ਹਰ ਪਰਿਮੇਯ ਪਦ ਨੂੰ LCD ਨਾਲ ਬਰਾਬਰ ਦੇ ਪਰਿਮੇਯ ਪਦ ਵਜੋਂ ਮੁੜ ਲਿਖੋ।
ਕਦਮ 2. ਪਰਿਮੇਯ ਪਦਾਂ ਨੂੰ ਜੋੜੋ ਜਾਂ ਘਟਾਓ।
ਕਦਮ 3. ਜੇ ਸੰਭਵ ਹੋਵੇ ਤਾਂ ਸਰਲ ਕਰੋ।
ਬਹੁਤ ਜਲਦੀ ਸਰਲ ਕਰਨ ਦੇ ਲਾਲਚ ਤੋਂ ਬਚੋ। ਉਪਰੋਕਤ ਉਦਾਹਰਨ ਵਿੱਚ, ਸਾਨੂੰ ਪਹਿਲੇ ਪਰਿਮੇਯ ਪਦ ਨੂੰ 3x−6(x−3)(x−2) ਵਜੋਂ ਛੱਡਣਾ ਪਵੇਗਾ ਤਾਂ ਜੋ ਇਸਨੂੰ 2x−6(x−2)(x−3) ਵਿੱਚ ਜੋੜਿਆ ਜਾ ਸਕੇ। ਸਿਰਫ ਅੰਸ਼ਾਂ ਨੂੰ ਜੋੜਨ ਤੋਂ ਬਾਅਦ ਹੀ ਸਰਲ ਕਰੋ।
ਉਦਾਹਰਨ 7.18
Add: 8x2−2x−3+3xx2+4x+3.8x2−2x−3+3xx2+4x+3.
Solution
row: Do the expressions have a common denominator? | No.
row: Rewrite each expression with the LCD.
row: Find the LCD.x2−2x−3=(x+1)(x−3)x2+4x+3=(x+1)(x+3)______________________________LCD=(x+1)(x−3)(x+3)Find the LCD.x2−2x−3=(x+1)(x−3)x2+4x+3=(x+1)(x+3)______________________________LCD=(x+1)(x−3)(x+3)
row: Rewrite each rational expression as anequivalent rational expression with the LCD.
row: Simplify the numerators.
row: Add the rational expressions.
row: Simplify the numerator.
row: The numerator is prime, so there areno common factors.
Try It 7.35
Add: 1m2−m−2+5mm2+3m+2.1m2−m−2+5mm2+3m+2.
Try It 7.36
Add:2nn2−3n−10+6n2+5n+6.2nn2−3n−10+6n2+5n+6.
The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.
Example 7.19
Subtract: 8yy2−16−4y−4.8yy2−16−4y−4.
Solution
row: Do the expressions have a common denominator? | No.
row: Rewrite each expression with the LCD.
row: Find the LCD.y2−16=(y−4)(y+4)y−4=y−4____________LCD=(y−4)(y+4)Find the LCD.y2−16=(y−4)(y+4)y−4=y−4____________LCD=(y−4)(y+4)
row: Rewrite each rational expression as anequivalent rational expression with the LCD.
row: Simplify the numerators.
row: Subtract the rational expressions.
row: Simplify the numerator.
row: Factor the numerator to look for common factors.
row: Remove common factors
row: Simplify.
Try It 7.37
Subtract: 2xx2−4−1x+2.2xx2−4−1x+2.
Try It 7.38
Subtract: 3z+3−6zz2−9.3z+3−6zz2−9.
There are lots of negative signs in the next example. Be extra careful.
Example 7.20
Subtract:−3n−9n2+n−6−n+32−n.−3n−9n2+n−6−n+32−n.
Solution
row: Factor the denominator.
row: Since n−2n−2 and 2−n2−n are opposites, we willmultiply the second rational expression by −1−1.−1−1.
row: Simplify. Remember, a−(−b)=a+b.a−(−b)=a+b.
row: Do the rational expressions have acommon denominator? No.
row: Find the LCD.n2+n−6=(n−2)(n+3)n−2=(n−2)_________________LCD=(n−2)(n+3)Find the LCD.n2+n−6=(n−2)(n+3)n−2=(n−2)_________________LCD=(n−2)(n+3)
row: Rewrite each rational expression as anequivalent rational expression with the LCD.
row: Simplify the numerators.
row: Add the rational expressions.
row: Simplify the numerator.
row: Factor the numerator to look for common factors.
row: Simplify.
Try It 7.39
Subtract :3x−1x2−5x−6−26−x.3x−1x2−5x−6−26−x.
Try It 7.40
Subtract: −2y−2y2+2y−8−y−12−y.−2y−2y2+2y−8−y−12−y.
Things can get very messy when both fractions must be multiplied by a binomial to get the common denominator.
Example 7.21
Subtract: 4a2+6a+5−3a2+7a+10.4a2+6a+5−3a2+7a+10.
Solution
row: Factor the denominators.
row: Do the rational expressions have acommon denominator? No.
row: Find the LCD.a2+6a+5=(a+1)(a+5)a2+7a+10=(a+5)(a+2)____________________________LCD=(a+1)(a+5)(a+2)Find the LCD.a2+6a+5=(a+1)(a+5)a2+7a+10=(a+5)(a+2)____________________________LCD=(a+1)(a+5)(a+2)
row: Rewrite each rational expression as anequivalent rational expression with the LCD.
row: Simplify the numerators.
row: Subtract the rational expressions.
row: Simplify the numerator.
row: Look for common factors.
row: Simplify.
Try It 7.41
Subtract: 3b2−4b−5−2b2−6b+5.3b2−4b−5−2b2−6b+5.
Try It 7.42
Subtract: 4x2−4−3x2−x−2.4x2−4−3x2−x−2.
We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example, we will start by factoring all three denominators to find their LCD.
Example 7.22
Simplify: 2uu−1+1u−2u−1u2−u.2uu−1+1u−2u−1u2−u.
Solution
row: Do the expressions have a common denominator? No.Rewrite each expression with the LCD.
row: Find the LCD. u−1=(u−1) u=u u2−u=u(u−1)_______________ LCD=u(u−1)Find the LCD. u−1=(u−1) u=u u2−u=u(u−1)_______________ LCD=u(u−1)
row: Rewrite each rational expression as anequivalent rational expression with the LCD.
row: Write as one rational expression.
row: Simplify.
row: Factor the numerator, and removecommon factors.
row: Simplify.
Try It 7.43
Simplify: vv+1+3v−1−6v2−1.vv+1+3v−1−6v2−1.
Try It 7.44
Simplify: 3ww+2+2w+7−17w+4w2+9w+14.3ww+2+2w+7−17w+4w2+9w+14.
Add and subtract rational functions
To add or subtract rational functions, we use the same techniques we used to add or subtract polynomial functions.
Example 7.23
Find R(x)=f(x)−g(x)R(x)=f(x)−g(x) where f(x)=x+5x−2f(x)=x+5x−2 and g(x)=5x+18x2−4.g(x)=5x+18x2−4.
Solution
row: Substitute in the functions f(x),f(x),g(x).g(x).
row: Factor the denominators.
row: Do the expressions have a common denominator? No.Rewrite each expression with the LCD.
row: Find the LCD. x−2=(x−2) x2−4=(x−2)(x+2)___________________ LCD=(x−2)(x+2)Find the LCD. x−2=(x−2) x2−4=(x−2)(x+2)___________________ LCD=(x−2)(x+2)
row: Rewrite each rational expression as anequivalent rational expression with the LCD.
row: Write as one rational expression.
row: Simplify.
row: Factor the numerator, and removecommon factors.
row: Simplify.
Try It 7.45
Find R(x)=f(x)−g(x)R(x)=f(x)−g(x) where f(x)=x+1x+3f(x)=x+1x+3 and g(x)=x+17x2−x−12.g(x)=x+17x2−x−12.
Try It 7.46
Find R(x)=f(x)+g(x)R(x)=f(x)+g(x) where f(x)=x−4x+3f(x)=x−4x+3 and g(x)=4x+6x2−9.g(x)=4x+6x2−9.
Media
Access this online resource for additional instruction and practice with adding and subtracting rational expressions.
Add and Subtract Rational Expressions- Unlike Denominators
Practice Makes Perfect
Add and Subtract Rational Expressions with a Common Denominator
In the following exercises, add.
2 15 + 7 15 2 15 + 7 15
7 24 + 11 24 7 24 + 11 24
3 c 4 c − 5 + 5 4 c − 5 3 c 4 c − 5 + 5 4 c − 5
7 m 2 m + n + 4 2 m + n 7 m 2 m + n + 4 2 m + n
2 r 2 2 r − 1 + 15 r − 8 2 r − 1 2 r 2 2 r − 1 + 15 r − 8 2 r − 1
3 s 2 3 s − 2 + 13 s − 10 3 s − 2 3 s 2 3 s − 2 + 13 s − 10 3 s − 2
2 w 2 w 2 − 16 + 8 w w 2 − 16 2 w 2 w 2 − 16 + 8 w w 2 − 16
7 x 2 x 2 − 9 + 21 x x 2 − 9 7 x 2 x 2 − 9 + 21 x x 2 − 9
In the following exercises, subtract.
9 a 2 3 a − 7 − 49 3 a − 7 9 a 2 3 a − 7 − 49 3 a − 7
25 b 2 5 b − 6 − 36 5 b − 6 25 b 2 5 b − 6 − 36 5 b − 6
3 m 2 6 m − 30 − 21 m − 30 6 m − 30 3 m 2 6 m − 30 − 21 m − 30 6 m − 30
2 n 2 4 n − 32 − 18 n − 16 4 n − 32 2 n 2 4 n − 32 − 18 n − 16 4 n − 32
6 p 2 + 3 p + 4 p 2 + 4 p − 5 − 5 p 2 + p + 7 p 2 + 4 p − 5 6 p 2 + 3 p + 4 p 2 + 4 p − 5 − 5 p 2 + p + 7 p 2 + 4 p − 5
5 q 2 + 3 q − 9 q 2 + 6 q + 8 − 4 q 2 + 9 q + 7 q 2 + 6 q + 8 5 q 2 + 3 q − 9 q 2 + 6 q + 8 − 4 q 2 + 9 q + 7 q 2 + 6 q + 8
5 r 2 + 7 r − 33 r 2 − 49 − 4 r 2 + 5 r + 30 r 2 − 49 5 r 2 + 7 r − 33 r 2 − 49 − 4 r 2 + 5 r + 30 r 2 − 49
7 t 2 − t − 4 t 2 − 25 − 6 t 2 + 12 t − 44 t 2 − 25 7 t 2 − t − 4 t 2 − 25 − 6 t 2 + 12 t − 44 t 2 − 25
Add and Subtract Rational Expressions whose Denominators are Opposites
In the following exercises, add or subtract.
10 v 2 v − 1 + 2 v + 4 1 − 2 v 10 v 2 v − 1 + 2 v + 4 1 − 2 v
20 w 5 w − 2 + 5 w + 6 2 − 5 w 20 w 5 w − 2 + 5 w + 6 2 − 5 w
10 x 2 + 16 x − 7 8 x − 3 + 2 x 2 + 3 x − 1 3 − 8 x 10 x 2 + 16 x − 7 8 x − 3 + 2 x 2 + 3 x − 1 3 − 8 x
6 y 2 + 2 y − 11 3 y − 7 + 3 y 2 − 3 y + 17 7 − 3 y 6 y 2 + 2 y − 11 3 y − 7 + 3 y 2 − 3 y + 17 7 − 3 y
z 2 + 6 z z 2 − 25 − 3 z + 20 25 − z 2 z 2 + 6 z z 2 − 25 − 3 z + 20 25 − z 2
a 2 + 3 a a 2 − 9 − 3 a − 27 9 − a 2 a 2 + 3 a a 2 − 9 − 3 a − 27 9 − a 2
2 b 2 + 30 b − 13 b 2 − 49 − 2 b 2 − 5 b − 8 49 − b 2 2 b 2 + 30 b − 13 b 2 − 49 − 2 b 2 − 5 b − 8 49 − b 2
c 2 + 5 c − 10 c 2 − 16 − c 2 − 8 c − 10 16 − c 2 c 2 + 5 c − 10 c 2 − 16 − c 2 − 8 c − 10 16 − c 2
Find the Least Common Denominator of Rational Expressions
In the following exercises, ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.
5 x 2 − 2 x − 8 , 2 x x 2 − x − 12 5 x 2 − 2 x − 8 , 2 x x 2 − x − 12
8 y 2 + 12 y + 35 , 3 y y 2 + y − 42 8 y 2 + 12 y + 35 , 3 y y 2 + y − 42
9 z 2 + 2 z − 8 , 4 z z 2 − 4 9 z 2 + 2 z − 8 , 4 z z 2 − 4
6 a 2 + 14 a + 45 , 5 a a 2 − 81 6 a 2 + 14 a + 45 , 5 a a 2 − 81
4 b 2 + 6 b + 9 , 2 b b 2 − 2 b − 15 4 b 2 + 6 b + 9 , 2 b b 2 − 2 b − 15
5 c 2 − 4 c + 4 , 3 c c 2 − 7 c + 10 5 c 2 − 4 c + 4 , 3 c c 2 − 7 c + 10
2 3 d 2 + 14 d − 5 , 5 d 3 d 2 − 19 d + 6 2 3 d 2 + 14 d − 5 , 5 d 3 d 2 − 19 d + 6
3 5 m 2 − 3 m − 2 , 6 m 5 m 2 + 17 m + 6 3 5 m 2 − 3 m − 2 , 6 m 5 m 2 + 17 m + 6
ਵੱਖਰੇ ਹਰਾਂ ਵਾਲੇ ਤਰਕਸੰਗਤ ਵਿਅੰਜਨਾਂ ਨੂੰ ਜੋੜਨਾ ਅਤੇ ਘਟਾਉਣਾ
ਹੇਠਾਂ ਦਿੱਤੇ ਅਭਿਆਸਾਂ ਵਿੱਚ, ਦਰਸਾਈਆਂ ਗਈਆਂ ਕਿਰਿਆਵਾਂ ਕਰੋ।
7/10 x²y + 4/15 xy²
1/12 a³b² + 5/9 a²b³
3r + 4 + 2r - 5
4s - 7 + 5s + 3
5/(3w - 2) + 2/(w + 1)
4/(2x + 5) + 2/(x - 1)
2y/(y + 3) + 3/(y - 1)
3z/(z - 2) + 1/(z + 5)
5b/(a²b - 2a²) + 2b/(b² - 4)
4c/(d + 3c) + 1/(d² - 9)
-3m/(3m - 3) + 5m/(m² + 3m - 4)
8/(4n + 4) + 6n/(n² - n - 2)
3r/(r² + 7r + 6) + 9r/(r² + 4r + 3)
2s/(s² + 2s - 8) + 4s/(s² + 3s - 10)
t/(t - 6) - (t - 2)/(t + 6)
(x - 3)/(x + 6) - x/(x + 3)
5a/(a + 3) - a/(a + 6)
3b/(b - 2) - (b - 6)/(b - 8)
6m + 6 - 12m/(m² - 36)
4n + 4 - 8n/(n² - 16)
-9p - 17/(p² - 4p - 21) - (p - 1)/(7 - p)
-13q - 8/(q² + 2q - 24) - (q - 2)/(4 - q)
−2 r − 16 r 2 + 6 r − 16 − 5 2 − r −2 r − 16 r 2 + 6 r − 16 − 5 2 − r
2 t − 30 t 2 + 6 t − 27 − 2 3 − t 2 t − 30 t 2 + 6 t − 27 − 2 3 − t
2 x + 7 10 x − 1 + 3 2 x + 7 10 x − 1 + 3
8 y − 4 5 y + 2 − 6 8 y − 4 5 y + 2 − 6
3 x 2 − 3 x − 4 − 2 x 2 − 5 x + 4 3 x 2 − 3 x − 4 − 2 x 2 − 5 x + 4
4 x 2 − 6 x + 5 − 3 x 2 − 7 x + 10 4 x 2 − 6 x + 5 − 3 x 2 − 7 x + 10
5 x 2 + 8 x − 9 − 4 x 2 + 10 x + 9 5 x 2 + 8 x − 9 − 4 x 2 + 10 x + 9
3 2 x 2 + 5 x + 2 − 1 2 x 2 + 3 x + 1 3 2 x 2 + 5 x + 2 − 1 2 x 2 + 3 x + 1
5 a a − 2 + 9 a − 2 a + 18 a 2 − 2 a 5 a a − 2 + 9 a − 2 a + 18 a 2 − 2 a
2 b b − 5 + 3 2 b − 2 b − 15 2 b 2 − 10 b 2 b b − 5 + 3 2 b − 2 b − 15 2 b 2 − 10 b
c c + 2 + 5 c − 2 − 10 c c 2 − 4 c c + 2 + 5 c − 2 − 10 c c 2 − 4
6 d d − 5 + 1 d + 4 − 7 d − 5 d 2 − d − 20 6 d d − 5 + 1 d + 4 − 7 d − 5 d 2 − d − 20
3 d d + 2 + 4 d − d + 8 d 2 + 2 d 3 d d + 2 + 4 d − d + 8 d 2 + 2 d
2 q q + 5 + 3 q − 3 − 13 q + 15 q 2 + 2 q − 15 2 q q + 5 + 3 q − 3 − 13 q + 15 q 2 + 2 q − 15
Add and Subtract Rational Functions
In the following exercises, find ⓐ R(x)=f(x)+g(x)R(x)=f(x)+g(x) ⓑ R(x)=f(x)−g(x).R(x)=f(x)−g(x).
f(x)=−5x−5x2+x−6f(x)=−5x−5x2+x−6 andg(x)=x+12−xg(x)=x+12−x
f(x)=−4x−24x2+x−30f(x)=−4x−24x2+x−30 andg(x)=x+75−xg(x)=x+75−x
f(x)=6xx2−64f(x)=6xx2−64 andg(x)=3x−8g(x)=3x−8
f(x)=5x+7f(x)=5x+7 andg(x)=10xx2−49g(x)=10xx2−49
Writing Exercises
Donald thinks that 3x+4x3x+4x is 72x.72x. Is Donald correct? Explain.
Explain how you find the Least Common Denominator of x2+5x+4x2+5x+4 and x2−16.x2−16.
Felipe thinks 1x+1y1x+1y is 2x+y.2x+y. ⓐ Choose numerical values for x and y and evaluate 1x+1y.1x+1y. ⓑ Evaluate 2x+y2x+y for the same values of x and y you used in part ⓐ. ⓒ Explain why Felipe is wrong. ⓓ Find the correct expression for 1x+1y.1x+1y.
1. 4n^2+6n+9−1n^2−9 ਦਾ ਸਰਲ ਰੂਪ ਅਤੇ ਸਾਰੇ ਕਦਮਾਂ ਦੀ ਵਿਆਖਿਆ। ਸਭ ਤੋਂ ਪਹਿਲਾਂ, ਅਸੀਂ ਸਮਾਨ ਪਦਾਂ ਨੂੰ ਇਕੱਠਾ ਕਰਾਂਗੇ। (4n^2 - 1n^2) + 6n + (9 - 9) ਇਹ ਬਣਦਾ ਹੈ: 3n^2 + 6n + 0 ਜਾਂ ਸਿਰਫ਼: 3n^2 + 6n
2. ਸਵੈ-ਜਾਂਚ
3. ⓐ ਇਸ ਭਾਗ ਦੇ ਉਦੇਸ਼ਾਂ ਵਿੱਚ ਤੁਹਾਡੀ ਮੁਹਾਰਤ ਦਾ ਮੁਲਾਂਕਣ ਕਰਨ ਲਈ ਅਭਿਆਸਾਂ ਨੂੰ ਪੂਰਾ ਕਰਨ ਤੋਂ ਬਾਅਦ, ਇਸ ਚੈਕਲਿਸਟ ਦੀ ਵਰਤੋਂ ਕਰੋ।
4. ⓑ ਇਸ ਚੈਕਲਿਸਟ ਦੀ ਸਮੀਖਿਆ ਕਰਨ ਤੋਂ ਬਾਅਦ, ਸਾਰੇ ਉਦੇਸ਼ਾਂ ਲਈ ਆਤਮ-ਵਿਸ਼ਵਾਸ ਪ੍ਰਾਪਤ ਕਰਨ ਲਈ ਤੁਸੀਂ ਕੀ ਕਰੋਗੇ?