lions, solve for the zeros of all of the denominators in the factored expression. In the case of division, both the numerator and denominator of the divisor must be used. Both are needed because the reciprocal of this expression is used in the calculation. CHECK Your Understanding 1. Simplify. State any restrictions on the variables. a) X 5 8 C ) ( x + 1) (x- 5) (x + 4) (x + 4) X 2(x - 5) b) 6xy 513 X d) 2- 2 2x + 1 X 6x + 3 8 5x 2. Simplify. State any restrictions on the variables. a) 2x c) 3x(x - 6) (x - 6) 3 5 (x + 2) (x - 7 ) (x + 2 ) b) x - 7 2x - 14 25 d) x - - x+1 10 x - 2 12 - 6x 3. Simplify. State any restrictions on the variables. ( x + 1 )2 ( x - 1 )2 a) x-+ 2x - 3 X x- + 4x + 3 2x + 10 x - 25 b) x2 - 4x + 4 x - 2PRACTISING 4. Simplify. State any restrictions on the variables. a) 2x2 X 21 9x 7 c ) 2x y 3 xy2 4 x 2 y b) 7a 14a2 d) 3a263 9a2b 3 5 2ab2 14a2 5. Simplify. State any restrictions on the variables. a ) 2 (x + 1 ) 3 X x - 1 2(x - 2) 12x-4 6(x + 1) c) 9x3 X 2 - x b) 3a - 6 a- 2 3 (m + 4 ) 2 5 (m + 4 ) at 2 at2 d) 2m + 1 7m + 14 6. Simplify. State any restrictions on the variables. a) (x + 1) (x - 3) 2 ( x + 2 ) (x + 2) 2 X (x - 3) (x + 3) b ) 2(n2 - 7n + 12) 5(n -4) n- - n - 6 n- - 4 C) 2x - x - 1 X - x- 6 X 6x2 - 5x + 1 8.x2 + 14x + 5 d) 912 - 4 . 97 + 12y + 4 4y - 12 18 - 6y 7. Simplify. State any restrictions on the variables. a) x - 5 xy + 42 X x2 + 2xy + yz x 2 + 3xy - 282 x- - 12 b) 2a2 - 12ab + 1862 4a2 - 12ab allane s a2 - 7ab + 1062 a2 - 7ab + 1062 c) 10x2 + 3xy - yz 6.x2 + 3xy 9x2 - 1/2 12x + 4y d) 15m + mn - 2n2 2n - 14m X 7m- - 8mn + n2 5 m2 + 7mn + 2n215m + mn - 2n2 X 7m2 - 8mn + n2 2n - 14m 5 m2 + 7mn + 2n2 8. Simplify. State any restrictions on the variables. x2 + x - 6 x(2x - 1) 2 (2x - 1) 2 X x2 - 4 x 2 + 2x - 3 9. Determine the area of the triangle in simplified form. State the restrictions. 5x + 35 x + 3 4x2 x2 - 16x + 63 ying and Dividing Rational Expressionson 10. An object has mass m = p+1 3p + 1 and density p = p2 - 1 gribbA 9p + 6p+10368 variables. Determine its volume v, where p = -. State the restrictions on any 11. Liz claims that if x = y, she can show that x + y = 0 by following these steps: Since x = verifiedduz bre anib x 2 = 12 I squared both sides of the equation. So x2 - 12 = 0. I rearranged terms in the equation. x2 - y- = O x - y I divided both sides by x - y . (x D) ( x ty ) I factored and simplified . x - y x ty = 0 Sarit says that's impossible because if x = 1, then y = 1, since x = y. Substituting into Liz's final equation, x + y = 0, gives 1 + 1 = 2,Determine its volume v, where p = - variables. 9p+ 6p +168 State the restrictions on any Liz claims that if x = y, she can show that x + y = 0 by following these steps: Since x = jennischual So x2 - 1 = 0. I squared both sides of the equation. x - y I rearranged terms in the equation. x - y I divided both sides by x - y . (x J) ( x ty ) = x - y I factored and simplified . xty=0 Sarit says that's impossible because if x = 1, then y = 1, since x = y. Substituting into Liz's final equation, x + y = 0, gives 1 + 1 = 2, not 0. Explain the error in Liz's reasoning. 12. a) Why do you usually factor all numerators and denominators before multiplying rational functions? b) Are there any exceptions to the rule in part (a)? Explain. Sam says that dividing two rational functions and multiplying the first function by the reciprocal of the second will produce the same function. Is this true? Explain.multiplying rational functions? cors and denominators before b) Are there any exceptions to the rule in part (a)? Explain. ) Sam says that dividing two rational functions and multiplying the first function by the reciprocal of the second will produce the same function. Is this true? Explain. Extending 13. Simplify. State any restrictions on the variables. m - mn m - n 6m2 + 11mn + 3n2 2m2 - mn - 6n2 Am2 - 7mn - 2n2 3 m 2 + 7 mn + 2n2 14. Newton's law of gravitation states that any two objects exert a gravitational force on each other due to their masses, F, = G-, where F is the gravitational force, G is a constant (the universal gravitational constant), m, and my are the masses of the objects, and r is the separation distance between the centres of objects. The mass of Mercury is 2.2 times greater than the mass of Pluto. Pluto is 102.1 times as far from the Sun as Mercury. How many times greater is the gravitational force between the Sun and Mercury than the gravitational force between the Sun and Pluto? Chapter 2 Equivalent Algebraic Expressions 123