ਸਿੱਖਣ ਦੇ ਉਦੇਸ਼
ਇਸ ਭਾਗ ਦੇ ਅੰਤ ਤੱਕ, ਤੁਸੀਂ ਯੋਗ ਹੋਵੋਗੇ:
ਤਾਰਕਿਕ ਵਿਅੰਜਨਾਂ ਨੂੰ ਗੁਣਾ ਕਰਨਾ
ਤਾਰਕਿਕ ਵਿਅੰਜਨਾਂ ਨੂੰ ਭਾਗ ਕਰਨਾ
ਤਿਆਰ ਰਹੋ 8.4
ਸ਼ੁਰੂ ਕਰਨ ਤੋਂ ਪਹਿਲਾਂ, ਇਸ ਤਿਆਰੀ ਕਵਿਜ਼ ਨੂੰ ਲਓ।
ਜੇਕਰ ਤੁਸੀਂ ਕੋਈ ਸਮੱਸਿਆ ਗੁਆਚਦੇ ਹੋ, ਤਾਂ ਸੂਚੀਬੱਧ ਭਾਗ 'ਤੇ ਵਾਪਸ ਜਾਓ ਅਤੇ ਸਮੱਗਰੀ ਦੀ ਸਮੀਖਿਆ ਕਰੋ।
ਗੁਣਾ ਕਰੋ: 1415·635.1415·635. ਜੇਕਰ ਤੁਸੀਂ ਇਹ ਸਮੱਸਿਆ ਗੁਆਚੀ ਹੈ, ਤਾਂ ਉਦਾਹਰਨ 1.68 ਦੀ ਸਮੀਖਿਆ ਕਰੋ।
ਤਿਆਰ ਰਹੋ 8.5
ਭਾਗ ਕਰੋ: 1415÷635.1415÷635. ਜੇਕਰ ਤੁਸੀਂ ਇਹ ਸਮੱਸਿਆ ਗੁਆਚੀ ਹੈ, ਤਾਂ ਉਦਾਹਰਨ 1.71 ਦੀ ਸਮੀਖਿਆ ਕਰੋ।
ਤਿਆਰ ਰਹੋ 8.6
ਪੂਰੀ ਤਰ੍ਹਾਂ ਗੁਣਨਖੰਡ ਬਣਾਓ: 2x2−98.2x2−98. ਜੇਕਰ ਤੁਸੀਂ ਇਹ ਸਮੱਸਿਆ ਗੁਆਚੀ ਹੈ, ਤਾਂ ਉਦਾਹਰਨ 7.62 ਦੀ ਸਮੀਖਿਆ ਕਰੋ।
ਤਿਆਰ ਰਹੋ 8.7
ਪੂਰੀ ਤਰ੍ਹਾਂ ਗੁਣਨਖੰਡ ਬਣਾਓ: 10n3+10.10n3+10. ਜੇਕਰ ਤੁਸੀਂ ਇਹ ਸਮੱਸਿਆ ਗੁਆਚੀ ਹੈ, ਤਾਂ ਉਦਾਹਰਨ 7.65 ਦੀ ਸਮੀਖਿਆ ਕਰੋ।
ਤਿਆਰ ਰਹੋ 8.8
ਪੂਰੀ ਤਰ੍ਹਾਂ ਗੁਣਨਖੰਡ ਬਣਾਓ: 10p2−25pq−15q2.10p2−25pq−15q2. ਜੇਕਰ ਤੁਸੀਂ ਇਹ ਸਮੱਸਿਆ ਗੁਆਚੀ ਹੈ, ਤਾਂ ਉਦਾਹਰਨ 7.68 ਦੀ ਸਮੀਖਿਆ ਕਰੋ।
ਤਾਰਕਿਕ ਵਿਅੰਜਨਾਂ ਨੂੰ ਗੁਣਾ ਕਰਨਾ
ਤਾਰਕਿਕ ਵਿਅੰਜਨਾਂ ਨੂੰ ਗੁਣਾ ਕਰਨ ਲਈ, ਅਸੀਂ ਉਹੀ ਕਰਦੇ ਹਾਂ ਜੋ ਅਸੀਂ ਸੰਖਿਆਤਮਕ ਭਿੰਨਾਂ ਨਾਲ ਕੀਤਾ ਸੀ। ਅਸੀਂ ਅੰਸ਼ਾਂ ਨੂੰ ਗੁਣਾ ਕਰਦੇ ਹਾਂ ਅਤੇ ਹਰਾਂ ਨੂੰ ਗੁਣਾ ਕਰਦੇ ਹਾਂ। ਫਿਰ, ਜੇਕਰ ਕੋਈ ਸਾਂਝੇ ਗੁਣਨਖੰਡ ਹਨ, ਤਾਂ ਅਸੀਂ ਨਤੀਜੇ ਨੂੰ ਸਰਲ ਬਣਾਉਣ ਲਈ ਉਨ੍ਹਾਂ ਨੂੰ ਹਟਾ ਦਿੰਦੇ ਹਾਂ।
ਤਾਰਕਿਕ ਵਿਅੰਜਨਾਂ ਦਾ ਗੁਣਾ
ਜੇਕਰ p,q,r,sp,q,r,s ਬਹੁਪਦ ਹਨ ਜਿੱਥੇ q≠0ands≠0q≠0ands≠0, ਤਾਂ
ਤਾਰਕਿਕ ਵਿਅੰਜਨਾਂ ਨੂੰ ਗੁਣਾ ਕਰਨ ਲਈ, ਅੰਸ਼ਾਂ ਨੂੰ ਗੁਣਾ ਕਰੋ ਅਤੇ ਹਰਾਂ ਨੂੰ ਗੁਣਾ ਕਰੋ।
ਅਸੀਂ ਪਹਿਲੀ ਉਦਾਹਰਨ ਸੰਖਿਆਤਮਕ ਭਿੰਨਾਂ ਨਾਲ ਕਰਾਂਗੇ ਤਾਂ ਜੋ ਸਾਨੂੰ ਯਾਦ ਆ ਜਾਵੇ ਕਿ ਅਸੀਂ ਵੇਰੀਏਬਲ ਤੋਂ ਬਿਨਾਂ ਭਿੰਨਾਂ ਨੂੰ ਕਿਵੇਂ ਗੁਣਾ ਕੀਤਾ ਸੀ।
ਉਦਾਹਰਨ 8.17
ਗੁਣਾ ਕਰੋ: 1028·815.1028·815.
Solution
row: Multiply the numerators and denominators.
row: Look for common factors, and then remove them.
row: Simplify.
Try It 8.33
Mulitply: 610·1512.610·1512.
Try It 8.34
Mulitply: 2015·68.2015·68.
Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, x≠0x≠0 and y≠0y≠0.
Example 8.18
Mulitply: 2x3y2·6xy3x2y.2x3y2·6xy3x2y.
Solution
row: Multiply.
row: Factor the numerator and denominator completely, and then remove common factors.
row: Simplify.
Try It 8.35
Mulitply: 3pqq2·5p2q6pq.3pqq2·5p2q6pq.
Try It 8.36
Mulitply: 6x3y7x2·2xy3x2y.6x3y7x2·2xy3x2y.
Example 8.19
How to Multiply Rational Expressions
Mulitply: 2xx2-7x+12·x2−96x2.2xx2-7x+12·x2−96x2.
Solution
Try It 8.37
Mulitply: 5xx2+5x+6·x2−410x.5xx2+5x+6·x2−410x.
Try It 8.38
Mulitply: 9x2x2+11x+30·x2−363x2.9x2x2+11x+30·x2−363x2.
How To
Multiply a rational expression.
Step 1. Factor each numerator and denominator completely.
Step 2. Multiply the numerators and denominators.
Step 3. Simplify by dividing out common factors.
Example 8.20
Multiply: n2−7nn2+2n+1·n+12n.n2−7nn2+2n+1·n+12n.
Solution
row: n2−7nn2+2n+1·n+12nn2−7nn2+2n+1·n+12n
row: Factor each numerator and denominator. | n(n−7)(n+1)(n+1)·n+12nn(n−7)(n+1)(n+1)·n+12n
row: Multiply the numerators and the denominators. | n(n−7)(n+1)(n+1)(n+1)2nn(n−7)(n+1)(n+1)(n+1)2n
row: Remove common factors. | n(n−7)(n+1)(n+1)(n+1)2nn(n−7)(n+1)(n+1)(n+1)2n
row: Simplify. | n−72(n+1)n−72(n+1)
Try It 8.39
Multiply: x2−25x2−3x−10·x+2x.x2−25x2−3x−10·x+2x.
Try It 8.40
Multiply: x2−4xx2+5x+6·x+2x.x2−4xx2+5x+6·x+2x.
Example 8.21
Multiply: 16−4x2x−12·x2−5x−6x2−16.16−4x2x−12·x2−5x−6x2−16.
Solution
row: 16−4x2x−12·x2−5x−6x2−1616−4x2x−12·x2−5x−6x2−16
row: Factor each numerator and denominator. | 4(4−x)2(x−6)·(x−6)(x+1)(x−4)(x+4)4(4−x)2(x−6)·(x−6)(x+1)(x−4)(x+4)
row: Multiply the numerators and the denominators. | 4(4−x)(x−6)(x+1)2(x−6)(x−4)(x+4)4(4−x)(x−6)(x+1)2(x−6)(x−4)(x+4)
row: Remove common factors. | (−1)2·2(4−x)(x−6)(x+1)2(x−6)(x−4)(x+4)(−1)2·2(4−x)(x−6)(x+1)2(x−6)(x−4)(x+4)
row: Simplify. | −2(x+1)(x+4)−2(x+1)(x+4)
Try It 8.41
Multiply: 12x−6x2x2+8x·x2+11x+24x2−4.12x−6x2x2+8x·x2+11x+24x2−4.
Try It 8.42
Multiply: 9v−3v29v+36·v2+7v+12v2−9.9v−3v29v+36·v2+7v+12v2−9.
Example 8.22
Multiply: 2x−6x2−8x+15·x2−252x+10.2x−6x2−8x+15·x2−252x+10.
Solution
row: Factor each numerator and denominator.
row: Multiply the numerators and denominators.
row: Remove common factors.
row: Simplify.
Try It 8.43
Multiply: 3a−21a2−9a+14·a2−43a+6.3a−21a2−9a+14·a2−43a+6.
Try It 8.44
Multiply: b2−bb2+9b−10·b2−100b2−10b.b2−bb2+9b−10·b2−100b2−10b.
Divide Rational Expressions
To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions.
Remember, the reciprocal of abab is baba. To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.
Division of Rational Expressions
If p,q,r,sp,q,r,s are polynomials where q≠0,r≠0,s≠0q≠0,r≠0,s≠0, then
To divide rational expressions multiply the first fraction by the reciprocal of the second.
Example 8.23
How to Divide Rational Expressions
Divide: x+96−x÷x2−81x−6.x+96−x÷x2−81x−6.
Solution
Try It 8.45
Divide: c+35−c÷c2−9c−5.c+35−c÷c2−9c−5.
Try It 8.46
Divide: 2−dd−4÷4−d24−d.2−dd−4÷4−d24−d.
How To
Divide rational expressions.
Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
Step 2. Factor the numerators and denominators completely.
Step 3. Multiply the numerators and denominators together.
Step 4. Simplify by dividing out common factors.
Example 8.24
Divide: 3n2n2−4n÷9n2−45nn2−7n+10.3n2n2−4n÷9n2−45nn2−7n+10.
Solution
row: Rewrite the division as the product of the first rational expression and the reciprocal of the second.
row: Factor the numerators and denominators and then multiply.
row: Simplify by dividing out common factors.
Try It 8.47
Divide: 2m2m2−8m÷8m2+24mm2+m−6.2m2m2−8m÷8m2+24mm2+m−6.
Try It 8.48
Divide: 15n23n2+33n÷5n−5n2+9n−22.15n23n2+33n÷5n−5n2+9n−22.
Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.
Example 8.25
Divide: 2x2+5x−12x2−16÷2x2−13x+15x2−8x+16.2x2+5x−12x2−16÷2x2−13x+15x2−8x+16.
Solution
row: 2x2+5x−12x2−16÷2x2−13x+15x2−8x+162x2+5x−12x2−16÷2x2−13x+15x2−8x+16
row: Rewrite the division as multiplication ofthe first expression by the reciprocal of the second. | 2x2+5x−12x2−16·x2−8x+162x2−13x+152x2+5x−12x2−16·x2−8x+162x2−13x+15
row: Factor the numerators and denominators and then multiply. | (2x−3)(x+4)(x−4)(x−4)(x−4)(x+4)(2x−3)(x−5)(2x−3)(x+4)(x−4)(x−4)(x−4)(x+4)(2x−3)(x−5)
row: Simplify by dividing out common factors. | (2x−3)(x+4)(x−4)(x−4)(x−4)(x+4)(2x−3)(x−5)(2x−3)(x+4)(x−4)(x−4)(x−4)(x+4)(2x−3)(x−5)
row: Simplify. | x−4x−5x−4x−5
Try It 8.49
Divide: 3a2−8a−3a2−25÷3a2−14a−5a2+10a+25.3a2−8a−3a2−25÷3a2−14a−5a2+10a+25.
Try It 8.50
Divide: 4b2+7b−21−b2÷4b2+15b−4b2−2b+1.4b2+7b−21−b2÷4b2+15b−4b2−2b+1.
Example 8.26
Divide: p3+q32p2+2pq+2q2÷p2−q26.p3+q32p2+2pq+2q2÷p2−q26.
Solution
row: p3+q32p2+2pq+2q2÷p2−q26p3+q32p2+2pq+2q2÷p2−q26
row: Rewrite the division as a multiplicationof the first expression times thereciprocal of the second. | p3+q32p2+2pq+2q2·6p2−q2p3+q32p2+2pq+2q2·6p2−q2
row: Factor the numerators and denominators and then multiply. | (p+q)(p2−pq+q2)62(p2+pq+q2)(p−q)(p+q)(p+q)(p2−pq+q2)62(p2+pq+q2)(p−q)(p+q)
row: Simplify by dividing out common factors. | (p+q)(p2−pq+q2)632(p2+pq+q2)(p−q)(p+q)(p+q)(p2−pq+q2)632(p2+pq+q2)(p−q)(p+q)
row: Simplify. | 3(p2−pq+q2)(p−q)(p2+pq+q2)3(p2−pq+q2)(p−q)(p2+pq+q2)
Try It 8.51
Divide: x3−83x2−6x+12÷x2−46.x3−83x2−6x+12÷x2−46.
Try It 8.52
Divide: 2z2z2−1÷z3−z2+zz3−1.2z2z2−1÷z3−z2+zz3−1.
Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide 35÷435÷4, we first write 4 as a fraction so that we can find its reciprocal.
We do the same thing when we divide rational expressions.
Example 8.27
Divide: a2−b23ab÷(a2+2ab+b2).a2−b23ab÷(a2+2ab+b2).
Solution
row: a2−b23ab÷(a2+2ab+b2)a2−b23ab÷(a2+2ab+b2)
row: Write the second expression as a fraction. | a2−b23ab÷a2+2ab+b21a2−b23ab÷a2+2ab+b21
row: Rewrite the division as the firstexpression times the reciprocal of thesecond expression. | a2−b23ab·1a2+2ab+b2a2−b23ab·1a2+2ab+b2
row: Factor the numerators and thedenominators, and then multiply. | (a−b)(a+b)·13ab·(a+b)(a+b)(a−b)(a+b)·13ab·(a+b)(a+b)
row: Simplify by dividing out common factors. | (a−b)(a+b)3ab·(a+b)(a+b)(a−b)(a+b)3ab·(a+b)(a+b)
row: Simplify. | (a−b)3ab(a+b)(a−b)3ab(a+b)
Try It 8.53
Divide: 2x2−14x−164÷(x2+2x+1).2x2−14x−164÷(x2+2x+1).
Try It 8.54
Divide: y2−6y+8y2−4y÷(3y2−12y).y2−6y+8y2−4y÷(3y2−12y).
Remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.
Example 8.28
Divide: 6x2−7x+24x−82x2−7x+3x2−5x+6.6x2−7x+24x−82x2−7x+3x2−5x+6.
Solution
row: 6x2−7x+24x−82x2−7x+3x2−5x+66x2−7x+24x−82x2−7x+3x2−5x+6
row: Rewrite with a division sign. | 6x2−7x+24x−8÷2x2−7x+3x2−5x+66x2−7x+24x−8÷2x2−7x+3x2−5x+6
row: Rewrite as product of first timesreciprocal of second. | 6x2−7x+24x−8·x2−5x+62x2−7x+36x2−7x+24x−8·x2−5x+62x2−7x+3
row: Factor the numerators and thedenominators, and then multiply. | (2x−1)(3x−2)(x−2)(x−3)4(x−2)(2x−1)(x−3)(2x−1)(3x−2)(x−2)(x−3)4(x−2)(2x−1)(x−3)
row: Simplify by dividing out common factors. | (2x−1)(3x−2)(x−2)(x−3)4(x−2)(2x−1)(x−3)(2x−1)(3x−2)(x−2)(x−3)4(x−2)(2x−1)(x−3)
row: Simplify. | 3x−243x−24
Try It 8.55
Divide: 3x2+7x+24x+243x2−14x−5x2+x−30.3x2+7x+24x+243x2−14x−5x2+x−30.
Try It 8.56
Divide: y2−362y2+11y−62y2−2y−608y−4.y2−362y2+11y−62y2−2y−608y−4.
If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then we factor and multiply.
Example 8.29
Divide: 3x−64x−4·x2+2x−3x2−3x−10÷2x+128x+16.3x−64x−4·x2+2x−3x2−3x−10÷2x+128x+16.
Solution
row: Rewrite the division as multiplication by the reciprocal.
row: Factor the numerators and the denominators, and then multiply.
row: Simplify by dividing out common factors.
row: Simplify.
Try It 8.57
Divide: 4m+43m−15·m2−3m−10m2−4m−32÷12m−366m−48.4m+43m−15·m2−3m−10m2−4m−32÷12m−366m−48.
Try It 8.58
Divide: 2n2+10nn−1÷n2+10n+24n2+8n−9·n+48n2+12n.2n2+10nn−1÷n2+10n+24n2+8n−9·n+48n2+12n.
Practice Makes Perfect
Multiply Rational Expressions
In the following exercises, multiply.
12 16 · 4 10 12 16 · 4 10
32 5 · 16 24 32 5 · 16 24
18 10 · 4 30 18 10 · 4 30
21 36 · 45 24 21 36 · 45 24
5 x 2 y 4 12 x y 3 · 6 x 2 20 y 2 5 x 2 y 4 12 x y 3 · 6 x 2 20 y 2
8 w 3 y 9 y 2 · 3 y 4 w 4 8 w 3 y 9 y 2 · 3 y 4 w 4
12 a 3 b b 2 · 2 a b 2 9 b 3 12 a 3 b b 2 · 2 a b 2 9 b 3
4 m n 2 5 n 3 · m n 3 8 m 2 n 2 4 m n 2 5 n 3 · m n 3 8 m 2 n 2
5 p 2 p 2 − 5 p − 36 · p 2 − 16 10 p 5 p 2 p 2 − 5 p − 36 · p 2 − 16 10 p
3 q 2 q 2 + q − 6 · q 2 − 9 9 q 3 q 2 q 2 + q − 6 · q 2 − 9 9 q
4 r r 2 − 3 r − 10 · r 2 − 25 8 r 2 4 r r 2 − 3 r − 10 · r 2 − 25 8 r 2
s s 2 − 9 s + 14 · s 2 − 49 7 s 2 s s 2 − 9 s + 14 · s 2 − 49 7 s 2
x 2 − 7 x x 2 + 6 x + 9 · x + 3 4 x x 2 − 7 x x 2 + 6 x + 9 · x + 3 4 x
2 y 2 − 10 y y 2 + 10 y + 25 · y + 5 6 y 2 y 2 − 10 y y 2 + 10 y + 25 · y + 5 6 y
z 2 + 3 z z 2 − 3 z − 4 · z − 4 z 2 z 2 + 3 z z 2 − 3 z − 4 · z − 4 z 2
2 a 2 + 8 a a 2 − 9 a + 20 · a − 5 a 2 2 a 2 + 8 a a 2 − 9 a + 20 · a − 5 a 2
28 − 4 b 3 b − 3 · b 2 + 8 b − 9 b 2 − 49 28 − 4 b 3 b − 3 · b 2 + 8 b − 9 b 2 − 49
18 c − 2 c 2 6 c + 30 · c 2 + 7 c + 10 c 2 − 81 18 c − 2 c 2 6 c + 30 · c 2 + 7 c + 10 c 2 − 81
35 d − 7 d 2 d 2 + 7 d · d 2 + 12 d + 35 d 2 − 25 35 d − 7 d 2 d 2 + 7 d · d 2 + 12 d + 35 d 2 − 25
72 m − 12 m 2 8 m + 32 · m 2 + 10 m + 24 m 2 − 36 72 m − 12 m 2 8 m + 32 · m 2 + 10 m + 24 m 2 − 36
4 n + 20 n 2 + n − 20 · n 2 − 16 4 n + 16 4 n + 20 n 2 + n − 20 · n 2 − 16 4 n + 16
6 p 2 − 6 p p 2 + 7 p − 18 · p 2 − 81 3 p 2 − 27 p 6 p 2 − 6 p p 2 + 7 p − 18 · p 2 − 81 3 p 2 − 27 p
q 2 − 2 q q 2 + 6 q − 16 · q 2 − 64 q 2 − 8 q q 2 − 2 q q 2 + 6 q − 16 · q 2 − 64 q 2 − 8 q
2 r 2 − 2 r r 2 + 4 r − 5 · r 2 − 25 2 r 2 − 10 r 2 r 2 − 2 r r 2 + 4 r − 5 · r 2 − 25 2 r 2 − 10 r
Divide Rational Expressions
In the following exercises, divide.
t − 6 3 − t ÷ t − 5 t 2 − 9 t − 6 3 − t ÷ t − 5 t 2 − 9
v − 5 11 − v ÷ v 2 − 25 v − 11 v − 5 11 − v ÷ v 2 − 25 v − 11
10 + w w − 8 ÷ 100 − w 2 8 − w 10 + w w − 8 ÷ 100 − w 2 8 − w
7 + x x − 6 ÷ 49 − x x + 6 2 7 + x x − 6 ÷ 49 − x x + 6 2
27 y 2 3 y − 21 ÷ 3 y 2 + 18 y 2 + 13 y + 42 27 y 2 3 y − 21 ÷ 3 y 2 + 18 y 2 + 13 y + 42
24 z 2 2 z − 8 ÷ 4 z − 28 z 2 − 11 z + 28 24 z 2 2 z − 8 ÷ 4 z − 28 z 2 − 11 z + 28
16 a 2 4 a + 36 ÷ 4 a 2 − 24 a a 2 + 4 a − 45 16 a 2 4 a + 36 ÷ 4 a 2 − 24 a a 2 + 4 a − 45
24 b 2 2 b − 4 ÷ 12 b 2 + 36 b b 2 − 11 b + 18 24 b 2 2 b − 4 ÷ 12 b 2 + 36 b b 2 − 11 b + 18
5 c 2 + 9 c − 2 c 2 − 4 ÷ 5 c 2 − 16 c + 3 c 2 + 4 c + 4 5 c 2 + 9 c − 2 c 2 − 4 ÷ 5 c 2 − 16 c + 3 c 2 + 4 c + 4
2 d 2 + d − 3 d 2 − 16 ÷ 2 d 2 − 9 d − 18 d 2 − 8 d + 16 2 d 2 + d − 3 d 2 − 16 ÷ 2 d 2 − 9 d − 18 d 2 − 8 d + 16
6 m 2 − 11 m − 2 9 − m 2 ÷ 6 m 2 + 25 m + 4 m 2 − 6 m + 9 6 m 2 − 11 m − 2 9 − m 2 ÷ 6 m 2 + 25 m + 4 m 2 − 6 m + 9
2 n 2 − 3 n − 14 25 − n 2 ÷ 2 n 2 − 13 n + 21 n 2 − 10 n + 25 2 n 2 − 3 n − 14 25 − n 2 ÷ 2 n 2 − 13 n + 21 n 2 − 10 n + 25
3 s 2 s 2 − 16 ÷ s 3 + 4 s 2 + 16 s s 3 − 64 3 s 2 s 2 − 16 ÷ s 3 + 4 s 2 + 16 s s 3 − 64
r 2 − 9 15 ÷ r 3 − 27 5 r 2 + 15 r + 45 r 2 − 9 15 ÷ r 3 − 27 5 r 2 + 15 r + 45
p 3 + q 3 3 p 2 + 3 p q + 3 q 2 ÷ p 2 − q 2 12 p 3 + q 3 3 p 2 + 3 p q + 3 q 2 ÷ p 2 − q 2 12
v 3 − 8 w 3 2 v 2 + 4 v w + 8 w 2 ÷ v 2 − 4 w 2 4 v 3 − 8 w 3 2 v 2 + 4 v w + 8 w 2 ÷ v 2 − 4 w 2 4
t 2 − 9 2 t ÷ ( t 2 − 6 t + 9 ) t 2 − 9 2 t ÷ ( t 2 − 6 t + 9 )
x 2 + 3 x − 10 4 x ÷ ( 2 x 2 + 20 x + 50 ) x 2 + 3 x − 10 4 x ÷ ( 2 x 2 + 20 x + 50 )
2 y 2 − 10 y z − 48 z 2 2 y − 1 ÷ ( 4 y 2 − 32 y z ) 2 y 2 − 10 y z − 48 z 2 2 y − 1 ÷ ( 4 y 2 − 32 y z )
2 m 2 − 98 n 2 2 m + 6 ÷ ( m 2 − 7 m n ) 2 m 2 − 98 n 2 2 m + 6 ÷ ( m 2 − 7 m n )
2 a 2 − a − 21 5 a + 20 a 2 + 7 a + 12 a 2 + 8 a + 16 2 a 2 − a − 21 5 a + 20 a 2 + 7 a + 12 a 2 + 8 a + 16
3 b 2 + 2 b − 8 12 b + 18 3 b 2 + 2 b − 8 2 b 2 − 7 b − 15 3 b 2 + 2 b − 8 12 b + 18 3 b 2 + 2 b − 8 2 b 2 − 7 b − 15
12 c 2 − 12 2 c 2 − 3 c + 1 4 c + 4 6 c 2 − 13 c + 5 12 c 2 − 12 2 c 2 − 3 c + 1 4 c + 4 6 c 2 − 13 c + 5
4 d 2 + 7 d − 2 35 d + 10 d 2 − 4 7 d 2 − 12 d − 4 4 d 2 + 7 d − 2 35 d + 10 d 2 − 4 7 d 2 − 12 d − 4
10 m 2 + 80 m 3 m − 9 · m 2 + 4 m − 21 m 2 − 9 m + 20 10 m 2 + 80 m 3 m − 9 · m 2 + 4 m − 21 m 2 − 9 m + 20 ÷ 5 m 2 + 10 m 2 m − 10 ÷ 5 m 2 + 10 m 2 m − 10
4 n 2 + 32 n 3 n + 2 · 3 n 2 − n − 2 n 2 + n − 30 4 n 2 + 32 n 3 n + 2 · 3 n 2 − n − 2 n 2 + n − 30 ÷ 108 n 2 − 24 n n + 6 ÷ 108 n 2 − 24 n n + 6
12 p 2 + 3 p p + 3 ÷ p 2 + 2 p − 63 p 2 − p − 12 12 p 2 + 3 p p + 3 ÷ p 2 + 2 p − 63 p 2 − p − 12 · p − 7 9 p 3 − 9 p 2 · p − 7 9 p 3 − 9 p 2
6 q + 3 9 q 2 − 9 q ÷ q 2 + 14 q + 33 q 2 + 4 q − 5 6 q + 3 9 q 2 − 9 q ÷ q 2 + 14 q + 33 q 2 + 4 q − 5 · 4 q 2 + 12 q 12 q + 6 · 4 q 2 + 12 q 12 q + 6
Everyday Math
Probability The director of large company is interviewing applicants for two identical jobs. If w=w= the number of women applicants and m=m= the number of men applicants, then the probability that two women are selected for the jobs is ww+m·w−1w+m−1.ww+m·w−1w+m−1.
ⓐ Simplify the probability by multiplying the two rational expressions.
ⓑ Find the probability that two women are selected when w=5w=5 and m=10m=10.
Area of a triangle The area of a triangle with base b and height h is bh2.bh2. If the triangle is stretched to make a new triangle with base and height three times as much as in the original triangle, the area is 9bh2.9bh2. Calculate how the area of the new triangle compares to the area of the original triangle by dividing 9bh29bh2 by bh2bh2.
Writing Exercises
ⓐ Multiply 74·91074·910 and explain all your steps.
ⓑ Multiply nn−3·9n+3nn−3·9n+3 and explain all your steps.
ⓒ Evaluate your answer to part (b) when n=7.n=7. Did you get the same answer you got in part (a)? Why or why not?
ⓐ Divide 245÷6245÷6 and explain all your steps.
ⓑ Divide x2−1x÷(x+1)x2−1x÷(x+1) and explain all your steps.
ⓒ Evaluate your answer to part (b) when x=5.x=5. Did you get the same answer you got in part (a)? Why or why not?
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?