1.5 Simplifying Rational Expressions
1.5 Simplifying Rational Expressions Canada officially has two national games, lacrosse and hockey. Lacrosse is thought to have originated with the Algonquin tribes in the St. Lawrence Valley. The game was very popular in the late nineteenth century and was at one time an Olympic sport. Canadian lacrosse teams won gold medals at the Summer Olympics in 1904 and 1908. There are two forms of lacrosse—box lacrosse, which is played indoors, and field lacrosse. When field lacrosse is played under international rules, the width of the rectangular field can be represented by x and the area of the field by the polynomial 2 2x − 10x. Thus, the length of the field can be represented by x 2x 2 – 10x 2x2 − 10x the expression . x This is an example of a rational expression, which is a quotient whose numerator and denominator are polynomials. The following are also rational expressions. 2 2 5y y−4 3 x+1 a +b x+2 7 x+3 y2 − 1 a2 − b2 I NVESTIGATE & I NQUIRE 1. a) Factor x from the expression for the area of a lacrosse field, 2x − 10x. b) Record the other factor and explain why it represents the length of the field. c) Describe how you could simplify the other expression for the length, 2
2x2 − 10x , to give the same expression as in part b). x 2. a) b)
The rectangle shown has a width of 2y and an area of 2y2 + 6y. Factor 2y from the expression for the area. Record the other factor and explain why it represents the length.
2y
2y 2 + 6y
1.5 Simplifying Rational Expressions • MHR 35
Use the width and the area to write a rational expression that represents the length. d) Describe how you could simplify the rational expression from part c) to give the same expression as in part b). c)
Use your results from questions 1 and 2 to write a rule for simplifying a rational expression in which the denominator is a monomial factor of the numerator. 3.
Use your rule to simplify each of the following. 6r4 − 3r3 + 6r2 4t2 + 8t 10m3 + 5m2 + 15m a) b) c) 4t 5m 3r2 4.
The expressions from question 1 represent the dimensions of a lacrosse field for both the women’s and the men’s games. a) For the women’s game, played 12-a-side, x represents 60 m. What are the dimensions of the field, in metres? b) For the men’s game, played 10-a-side, x represents 55 m. What are the dimensions of the field, in metres? 5.
EXAMPLE 1 Monomial Denominator 24x3 + 6x2 + 12x Simplify . State the restriction on the variable. 6x SOLUTION
Factor the numerator:
3 2 24x + 6x + 12x 6x 2 6x(4x + x + 2) = 6x
1
Divide by the common factor, 6x:
6x (4x + x + 2) = 6x 2
1
2 = 4x + x + 2
Division by 0 is not defined, so exclude values of x for which 6x = 0. 6x = 0 when x = 0, so x ≠ 0 3 2 24x + 6x + 12x 2 Therefore, = 4x + x + 2, x ≠ 0 6x 36 MHR • Chapter 1
Excluded values are known as restrictions on the variable.
The solution to Example 1 could have been found by another method, since the distributive property also applies to division. 3 2+1 For example, = 7 7 2 1 =+ 7 7 24x3 + 6x2 + 12x 24x3 6x2 12x So, = + + 6x 6x 6x 6x = 4x2 + x + 2, x ≠ 0 EXAMPLE 2 Binomial Denominator x Express in simplest form. State the restrictions on the variable. 2 2x − 4x SOLUTION
Factor the denominator:
x 2 2x − 4x x = 2x(x − 2) 1
Divide by the common factor, x:
x = 2x(x − 2) 1
1 = 2(x − 2) 2 Exclude values of x for which 2x − 4x = 0. 2x2 − 4x = 2x(x − 2), so 2x2 − 4x = 0 when 2x(x − 2) = 0 2x = 0 or x − 2 = 0 x = 0 or x = 2 x 1 Therefore, = , x ≠ 0, 2. 2 2x − 4x 2(x − 2)
1.5 Simplifying Rational Expressions • MHR 37
EXAMPLE 3 Removing a Common Factor of –1 3 − 2x Simplify . State any restrictions on the variable. 4x − 6 SOLUTION
Factor the denominator: Factor –1 from the numerator:
3 − 2x 4x − 6 3 − 2x = 2(2x − 3) –1(2x − 3) = 2(2x − 3) 1
–1(2x − 3) Divide by the common factor, (2x – 3): = 2(2x − 3) 1
Exclude values of x for which 4x − 6 = 0. 4x − 6 = 0 when 2(2x − 3) = 0. 2x − 3 = 0 3 x= 2 3 − 2x 1 3 Therefore, = – , x ≠ . 4x − 6 2 2
1 –1 = or − 2 2
EXAMPLE 4 Trinomial Numerator and Denominator x2 + 3x − 10 Express 2 in simplest form. State the restrictions on the variable. x + 8x + 15 SOLUTION
2 x + 3x − 10 x2 + 8x + 15 (x + 5)(x − 2) Factor the numerator and the denominator: = (x + 5)(x + 3)
1
Divide by the common factor, (x + 5):
(x + 5)(x − 2) = (x + 5)(x + 3) 1
x−2 = x+3 38 MHR • Chapter 1
Exclude values of x for which x2 + 8x + 15 = 0. x2 + 8x + 15 = (x + 5)(x + 3), so x2 + 8x + 15 = 0 when (x + 5)(x + 3) = 0 x + 5 = 0 or x + 3 = 0 x = −5 or x = −3 x2 + 3x − 10 x − 2 Therefore, = , x ≠ −5, −3. x2 + 8x + 15 x + 3 EXAMPLE 5 Trinomial Numerator and Denominator 2y2 − y − 15 Simplify . State the restrictions on the variable. 4y2 − 13y + 3 SOLUTION 2y − y − 15 4y2 − 13y + 3 (y − 3)(2y + 5) = (4y − 1)(y − 3) 2
Factor the numerator and denominator:
1
Divide by the common factor, ( y – 3):
(y − 3) (2y + 5) = (4y − 1)(y − 3) 1
2y + 5 = 4y − 1 2 Exclude values of y for which 4y − 13y + 3 = 0. 4y2 − 13y + 3 = (4y − 1)(y − 3), so 4y2 − 13y + 3 = 0 when (4y − 1)( y − 3) = 0 4y − 1 = 0 or y − 3 = 0 1 y = or y = 3 4 2 2y − y − 15 2y + 5 1 Therefore, = , y ≠ , 3. 2 4 4y − 13y + 3 4y − 1 Key
Concepts
• To simplify rational expressions, a) factor the numerator and the denominator b) divide by common factors • To state the restriction(s) on the variable in a rational expression, determine and exclude the value(s) of the variable that make the denominator 0. 1.5 Simplifying Rational Expressions • MHR 39
Communicate
Yo u r
Understanding x+4 1. Explain why x ≠ 3 is a restriction on the variable for the expression . x−3 2 x −x 2. Describe how you would simplify . x 2 x + 3x + 2 Describe how you would simplify . x2 − x − 2 b) Describe how you would determine the restrictions on the variable. 4. Write an expression in one variable for the denominator of a rational expression, if the restrictions on the variable are x ≠ 2, −3.
3. a)
Practise In each of the following, state any restrictions on the variables.
A 1. Simplify. 2 3t3 + 6t2 − 15t 6a + 9a a) b) 3t 12a2 4 3 2 10y + 5y − 15y c) 5y 4 3 2 14n − 4n + 6n + 8n d) 2n2 –6x2y3 4m2 − 8mn e) f) 4mn –18x3y –4x4y2z 16a2bc g) h) 4a2b2c2 20x3y3z 21m(m − 4) i) 7m2 Express in simplest form. 2 5x 8t (t + 5) a) b) 5(x + 4) 4t(t − 5) 7x(x − 3) (m − 1)(m + 2) c) d) 2 14x (x − 3) (m + 4)(m − 1) 2 y 2x e) f) 2x + 8 y2 + 2y 2.
40 MHR • Chapter 1
10x 2 5x − 15x 3x y i) 2 6x y − 12xy2 g)
Simplify. 6t − 36 a) t−6 5x − 10 c) 3x − 6 8x2 + 4x e) 6x2 + 3x 4x + 4y g) 5x + 5y 5x y + 10x i) 2y2 + 4y
h)
4x 3 16x − 12x
3.
4m + 24 8m − 24 2 a + 2a d) a2 − 3a 2 2x − 2x f) 2x2 + 2x 2 4a b + 8ab h) 6a2 − 6a
b)
Express in simplest equivalent form. y2 + 10y + 25 m−2 a) b) m2 − 5m + 6 y+5 2 2x + 6 r −4 c) d) 2 x − 6x − 27 5r + 10 4.
2 a +a 2 a + 2a + 1 2w + 2 g) 2 2w + 3w + 1 2 8z + 6z i) 9z2 − 16
e)
Simplify. y−2 a) 2−y 2t − 1 c) 4 − 8t 2 x −1 e) 2 1−x
2 x −9 2 2x y − 6x y 2 3t − 8t + 4 h) 6t2 − 4t 2 2 5x + 3x y − 2y j) 3x2 + 3xy
f)
5.
3−x x−3 6 − 10w d) 15w − 9 1 − 4y2 f) 8y2 − 2 b)
Simplify. x2 + 4x + 4 a) x2 + 5x + 6 m2 − 5m + 6 c) m2 + 2m − 15 x2 − 10x + 24 e) x2 − 12x + 36 2 p + 8p + 16 g) p2 − 16 6v2 + 11v + 3 i) 4v2 + 8v + 3 3z2 − 7z + 2 k) 9z2 − 6z + 1 6.
2 a − a − 12 a2 − 9a + 20 y2 − 8y + 15 d) y2 − 25 2 n −n−2 f) n2 + n − 6 2 2t − t − 1 h) t2 − 3t + 2 2 6x − 13x + 6 j) 8x2 − 6x − 9 2m2 − mn − n2 l) 4m2 − 4mn − 3n2
b)
Apply, Solve, Communicate The area of a Saskatchewan flag can be represented by the polynomial x2 + 3x + 2 and its width by x + 1. a) Write a rational expression that represents the length. b) Write the expression in simplest form. c) If x represents 1 unit of length, what is the ratio length:width for a Saskatchewan flag? 7. Saskatchewan flag
B Simplify, if possible. 1−x x−1 a) b) x−1 x+1 2 2 3t − 7 t −s d) e) 2 3t − 7 (s + t) 8.
y +1 c) y2 − 1 3 2 x − 2x + 3x f) 2x2 − 4x + 6 2
For which values of x are the following rational expressions not defined? 2x − y 3 4x a) b) c) 3 x−y 3x + y x 2 2 2 3x + 5xy + 2y2 x x + 3x − 11 d) e) f) x3 − 8 x2 − 1 4x2 − 9y2 9.
1.5 Simplifying Rational Expressions • MHR 41
State whether each of the following rational x+1 expressions is equivalent to the expression . Explain. x−1 2 x +x x+2 4 + 4x a) b) c) 2 x−2 4x − 4 x −x 3x + 1 (x + 1)2 1+x d) e) 2 f) 3x − 1 1−x (x − 1) 10. Communication
For a cube of edge length x, find the ratio of the volume to the surface area. Simplify, if possible.
11. Cube
For a sphere of radius r, find the ratio of the volume to the surface area. Simplify, if possible. 12. Application 13. Pattern
The first 4 diagrams of two patterns are shown.
Pattern 1
n=1
2
Pattern 2
3
4
n=1
2
3
4
For pattern 1, express the number of asterisks in the nth diagram in terms of n. b) For pattern 2, the number of asterisks in the nth diagram is given by the binomial product (n + ▲)(n + ■ ), where ▲ and ■ represent whole numbers. Replace ▲ and ■ in the binomial product by their correct values. c) Divide your polynomial from part b) by your expression from part a). d) Use your result from part c) to calculate how many times as many asterisks there are in the 10th diagram of pattern 2 as there are in the 10th diagram of pattern 1. e) If a diagram in pattern 1 has 20 asterisks, how many asterisks are in the corresponding diagram of pattern 2? f) If a diagram in pattern 2 has 1295 asterisks, how many asterisks are in the corresponding diagram in pattern 1? a)
42 MHR • Chapter 1
Find the ratio of the area of the square to the area of the trapezoid. Simplify, if possible. 14. Measurement
10x
8
8
Find the ratio of the volume to the surface area for the rectangular prism shown. Simplify, if possible. 15. Rectangular prism
x+4
2x – 2 x+4
C
Write rational expressions in one variable so that the restrictions on the variables are as follows. 1 3 a) x ≠ 1 b) y ≠ 0, −3 c) a ≠ , – d) t ≠ −1, ±3 2 4 16.
17. Technology a)
Use a graphing calculator to graph the equations
2x + 3x y = and y = 2x + 3 in the same standard viewing window. x Explain your observations. b) Display the tables of values for the two equations. Compare and explain the values of y when x = 0. 2
For a solid cone with radius r, height h, and slant height s, find the ratio of the volume to the surface area. Simplify, if possible. b) Determine whole-number values of r, h, and s that give the ratio in part a) a numerical value of 1. 18. Inquiry/Problem Solving a)
WORD
Power
Lewis Carroll invented a word game called doublets. The object of the game is to change one word to another by changing one letter at a time. You must form a real word each time you change a letter. The best solution has the fewest steps. Change the word RING to the word BELL by changing one letter at a time. 1.5 Simplifying Rational Expressions • MHR 43
2x2 − 10x , to give the same expression as in part b). x 2. a) b)
The rectangle shown has a width of 2y and an area of 2y2 + 6y. Factor 2y from the expression for the area. Record the other factor and explain why it represents the length.
2y
2y 2 + 6y
1.5 Simplifying Rational Expressions • MHR 35
Use the width and the area to write a rational expression that represents the length. d) Describe how you could simplify the rational expression from part c) to give the same expression as in part b). c)
Use your results from questions 1 and 2 to write a rule for simplifying a rational expression in which the denominator is a monomial factor of the numerator. 3.
Use your rule to simplify each of the following. 6r4 − 3r3 + 6r2 4t2 + 8t 10m3 + 5m2 + 15m a) b) c) 4t 5m 3r2 4.
The expressions from question 1 represent the dimensions of a lacrosse field for both the women’s and the men’s games. a) For the women’s game, played 12-a-side, x represents 60 m. What are the dimensions of the field, in metres? b) For the men’s game, played 10-a-side, x represents 55 m. What are the dimensions of the field, in metres? 5.
EXAMPLE 1 Monomial Denominator 24x3 + 6x2 + 12x Simplify . State the restriction on the variable. 6x SOLUTION
Factor the numerator:
3 2 24x + 6x + 12x 6x 2 6x(4x + x + 2) = 6x
1
Divide by the common factor, 6x:
6x (4x + x + 2) = 6x 2
1
2 = 4x + x + 2
Division by 0 is not defined, so exclude values of x for which 6x = 0. 6x = 0 when x = 0, so x ≠ 0 3 2 24x + 6x + 12x 2 Therefore, = 4x + x + 2, x ≠ 0 6x 36 MHR • Chapter 1
Excluded values are known as restrictions on the variable.
The solution to Example 1 could have been found by another method, since the distributive property also applies to division. 3 2+1 For example, = 7 7 2 1 =+ 7 7 24x3 + 6x2 + 12x 24x3 6x2 12x So, = + + 6x 6x 6x 6x = 4x2 + x + 2, x ≠ 0 EXAMPLE 2 Binomial Denominator x Express in simplest form. State the restrictions on the variable. 2 2x − 4x SOLUTION
Factor the denominator:
x 2 2x − 4x x = 2x(x − 2) 1
Divide by the common factor, x:
x = 2x(x − 2) 1
1 = 2(x − 2) 2 Exclude values of x for which 2x − 4x = 0. 2x2 − 4x = 2x(x − 2), so 2x2 − 4x = 0 when 2x(x − 2) = 0 2x = 0 or x − 2 = 0 x = 0 or x = 2 x 1 Therefore, = , x ≠ 0, 2. 2 2x − 4x 2(x − 2)
1.5 Simplifying Rational Expressions • MHR 37
EXAMPLE 3 Removing a Common Factor of –1 3 − 2x Simplify . State any restrictions on the variable. 4x − 6 SOLUTION
Factor the denominator: Factor –1 from the numerator:
3 − 2x 4x − 6 3 − 2x = 2(2x − 3) –1(2x − 3) = 2(2x − 3) 1
–1(2x − 3) Divide by the common factor, (2x – 3): = 2(2x − 3) 1
Exclude values of x for which 4x − 6 = 0. 4x − 6 = 0 when 2(2x − 3) = 0. 2x − 3 = 0 3 x= 2 3 − 2x 1 3 Therefore, = – , x ≠ . 4x − 6 2 2
1 –1 = or − 2 2
EXAMPLE 4 Trinomial Numerator and Denominator x2 + 3x − 10 Express 2 in simplest form. State the restrictions on the variable. x + 8x + 15 SOLUTION
2 x + 3x − 10 x2 + 8x + 15 (x + 5)(x − 2) Factor the numerator and the denominator: = (x + 5)(x + 3)
1
Divide by the common factor, (x + 5):
(x + 5)(x − 2) = (x + 5)(x + 3) 1
x−2 = x+3 38 MHR • Chapter 1
Exclude values of x for which x2 + 8x + 15 = 0. x2 + 8x + 15 = (x + 5)(x + 3), so x2 + 8x + 15 = 0 when (x + 5)(x + 3) = 0 x + 5 = 0 or x + 3 = 0 x = −5 or x = −3 x2 + 3x − 10 x − 2 Therefore, = , x ≠ −5, −3. x2 + 8x + 15 x + 3 EXAMPLE 5 Trinomial Numerator and Denominator 2y2 − y − 15 Simplify . State the restrictions on the variable. 4y2 − 13y + 3 SOLUTION 2y − y − 15 4y2 − 13y + 3 (y − 3)(2y + 5) = (4y − 1)(y − 3) 2
Factor the numerator and denominator:
1
Divide by the common factor, ( y – 3):
(y − 3) (2y + 5) = (4y − 1)(y − 3) 1
2y + 5 = 4y − 1 2 Exclude values of y for which 4y − 13y + 3 = 0. 4y2 − 13y + 3 = (4y − 1)(y − 3), so 4y2 − 13y + 3 = 0 when (4y − 1)( y − 3) = 0 4y − 1 = 0 or y − 3 = 0 1 y = or y = 3 4 2 2y − y − 15 2y + 5 1 Therefore, = , y ≠ , 3. 2 4 4y − 13y + 3 4y − 1 Key
Concepts
• To simplify rational expressions, a) factor the numerator and the denominator b) divide by common factors • To state the restriction(s) on the variable in a rational expression, determine and exclude the value(s) of the variable that make the denominator 0. 1.5 Simplifying Rational Expressions • MHR 39
Communicate
Yo u r
Understanding x+4 1. Explain why x ≠ 3 is a restriction on the variable for the expression . x−3 2 x −x 2. Describe how you would simplify . x 2 x + 3x + 2 Describe how you would simplify . x2 − x − 2 b) Describe how you would determine the restrictions on the variable. 4. Write an expression in one variable for the denominator of a rational expression, if the restrictions on the variable are x ≠ 2, −3.
3. a)
Practise In each of the following, state any restrictions on the variables.
A 1. Simplify. 2 3t3 + 6t2 − 15t 6a + 9a a) b) 3t 12a2 4 3 2 10y + 5y − 15y c) 5y 4 3 2 14n − 4n + 6n + 8n d) 2n2 –6x2y3 4m2 − 8mn e) f) 4mn –18x3y –4x4y2z 16a2bc g) h) 4a2b2c2 20x3y3z 21m(m − 4) i) 7m2 Express in simplest form. 2 5x 8t (t + 5) a) b) 5(x + 4) 4t(t − 5) 7x(x − 3) (m − 1)(m + 2) c) d) 2 14x (x − 3) (m + 4)(m − 1) 2 y 2x e) f) 2x + 8 y2 + 2y 2.
40 MHR • Chapter 1
10x 2 5x − 15x 3x y i) 2 6x y − 12xy2 g)
Simplify. 6t − 36 a) t−6 5x − 10 c) 3x − 6 8x2 + 4x e) 6x2 + 3x 4x + 4y g) 5x + 5y 5x y + 10x i) 2y2 + 4y
h)
4x 3 16x − 12x
3.
4m + 24 8m − 24 2 a + 2a d) a2 − 3a 2 2x − 2x f) 2x2 + 2x 2 4a b + 8ab h) 6a2 − 6a
b)
Express in simplest equivalent form. y2 + 10y + 25 m−2 a) b) m2 − 5m + 6 y+5 2 2x + 6 r −4 c) d) 2 x − 6x − 27 5r + 10 4.
2 a +a 2 a + 2a + 1 2w + 2 g) 2 2w + 3w + 1 2 8z + 6z i) 9z2 − 16
e)
Simplify. y−2 a) 2−y 2t − 1 c) 4 − 8t 2 x −1 e) 2 1−x
2 x −9 2 2x y − 6x y 2 3t − 8t + 4 h) 6t2 − 4t 2 2 5x + 3x y − 2y j) 3x2 + 3xy
f)
5.
3−x x−3 6 − 10w d) 15w − 9 1 − 4y2 f) 8y2 − 2 b)
Simplify. x2 + 4x + 4 a) x2 + 5x + 6 m2 − 5m + 6 c) m2 + 2m − 15 x2 − 10x + 24 e) x2 − 12x + 36 2 p + 8p + 16 g) p2 − 16 6v2 + 11v + 3 i) 4v2 + 8v + 3 3z2 − 7z + 2 k) 9z2 − 6z + 1 6.
2 a − a − 12 a2 − 9a + 20 y2 − 8y + 15 d) y2 − 25 2 n −n−2 f) n2 + n − 6 2 2t − t − 1 h) t2 − 3t + 2 2 6x − 13x + 6 j) 8x2 − 6x − 9 2m2 − mn − n2 l) 4m2 − 4mn − 3n2
b)
Apply, Solve, Communicate The area of a Saskatchewan flag can be represented by the polynomial x2 + 3x + 2 and its width by x + 1. a) Write a rational expression that represents the length. b) Write the expression in simplest form. c) If x represents 1 unit of length, what is the ratio length:width for a Saskatchewan flag? 7. Saskatchewan flag
B Simplify, if possible. 1−x x−1 a) b) x−1 x+1 2 2 3t − 7 t −s d) e) 2 3t − 7 (s + t) 8.
y +1 c) y2 − 1 3 2 x − 2x + 3x f) 2x2 − 4x + 6 2
For which values of x are the following rational expressions not defined? 2x − y 3 4x a) b) c) 3 x−y 3x + y x 2 2 2 3x + 5xy + 2y2 x x + 3x − 11 d) e) f) x3 − 8 x2 − 1 4x2 − 9y2 9.
1.5 Simplifying Rational Expressions • MHR 41
State whether each of the following rational x+1 expressions is equivalent to the expression . Explain. x−1 2 x +x x+2 4 + 4x a) b) c) 2 x−2 4x − 4 x −x 3x + 1 (x + 1)2 1+x d) e) 2 f) 3x − 1 1−x (x − 1) 10. Communication
For a cube of edge length x, find the ratio of the volume to the surface area. Simplify, if possible.
11. Cube
For a sphere of radius r, find the ratio of the volume to the surface area. Simplify, if possible. 12. Application 13. Pattern
The first 4 diagrams of two patterns are shown.
Pattern 1
n=1
2
Pattern 2
3
4
n=1
2
3
4
For pattern 1, express the number of asterisks in the nth diagram in terms of n. b) For pattern 2, the number of asterisks in the nth diagram is given by the binomial product (n + ▲)(n + ■ ), where ▲ and ■ represent whole numbers. Replace ▲ and ■ in the binomial product by their correct values. c) Divide your polynomial from part b) by your expression from part a). d) Use your result from part c) to calculate how many times as many asterisks there are in the 10th diagram of pattern 2 as there are in the 10th diagram of pattern 1. e) If a diagram in pattern 1 has 20 asterisks, how many asterisks are in the corresponding diagram of pattern 2? f) If a diagram in pattern 2 has 1295 asterisks, how many asterisks are in the corresponding diagram in pattern 1? a)
42 MHR • Chapter 1
Find the ratio of the area of the square to the area of the trapezoid. Simplify, if possible. 14. Measurement
10x
8
8
Find the ratio of the volume to the surface area for the rectangular prism shown. Simplify, if possible. 15. Rectangular prism
x+4
2x – 2 x+4
C
Write rational expressions in one variable so that the restrictions on the variables are as follows. 1 3 a) x ≠ 1 b) y ≠ 0, −3 c) a ≠ , – d) t ≠ −1, ±3 2 4 16.
17. Technology a)
Use a graphing calculator to graph the equations
2x + 3x y = and y = 2x + 3 in the same standard viewing window. x Explain your observations. b) Display the tables of values for the two equations. Compare and explain the values of y when x = 0. 2
For a solid cone with radius r, height h, and slant height s, find the ratio of the volume to the surface area. Simplify, if possible. b) Determine whole-number values of r, h, and s that give the ratio in part a) a numerical value of 1. 18. Inquiry/Problem Solving a)
WORD
Power
Lewis Carroll invented a word game called doublets. The object of the game is to change one word to another by changing one letter at a time. You must form a real word each time you change a letter. The best solution has the fewest steps. Change the word RING to the word BELL by changing one letter at a time. 1.5 Simplifying Rational Expressions • MHR 43
