By Deborah Hughes-Hallett, William G. McCallum, Eric Connally
Shape is expounded to operate. An aircraft wing has the shape it does as a result of its lifting functionality. The pillars of the Parthenon and the girders of a skyscraper are formed to the aim of assisting their vast constructions. equally, the shape of an algebraic expression or equation displays its functionality. Algebra: shape and serve as initial version introduces each one function--linear, energy, quadratic, exponential, polynomial--and provides a examine of the elemental kind of expressions for that functionality. Readers are inspired to envision the elemental varieties, see how they're built, and look at the position of every part. during the textual content, there are instruments sections positioned on the ends of chapters to aid readers collect the talents they should practice uncomplicated algebraic manipulations.
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Extra info for Algebra: Form and Function
74. In Problems 37–41, complete the table. Which, if any, of the expressions in the left-hand column are equivalent to each other? Justify your answer algebraically. 0 −1 m4 4m2 40. I −2 −1 0 1 2 −I −(−I) x −11 −7 0 7 11 2x 3x 41. x 2x + 3x −2 −1 0 1 2 x+3 5x −(x + 3) 38. −x t −11 −7 0 7 11 −x + 3 2t −x − 3 −3t 2t − 3t 42. Write a sentence explaining what it means for two expressions to be equivalent. 2, the equations were simple enough to solve by direct reasoning about numbers. But how can we be sure of finding a solution to an equation, and how can we be sure that we have found all the solutions?
A) 6(s + 3w) (b) 4(s − 3t + 4w) (c) a(b + c − d) (a) Here we have 6(s + 3w) = 6s + 18w. (b) Distributing over the three terms in the sum, we have 4(s − 3t + 4w) = 4s − 12t + 16w. (c) Again, distributing over three terms, we get a(b + c − d) = ab + ac − ad. Distributing a Negative Sign Since −x = (−1)x, the distributive law tells us how to distribute a negative sign: −(b + c) = (−1)(b + c) a = (−1)b + (−1)c by the distributive law ab ac = −b − c. Example 3 Express −(c − d) − (d + c) in a simpler way.
X/5 and 5/x 4. 2x2 − 5x and 25 − 3x2 In Exercises 5–14, are the expressions equivalent? 5. 3 EQUIVALENT EXPRESSIONS 6. 5r 7. 3(z + w) and 3z + 3w √ 9. a2 + b2 and a + b 11. (3 − 4t)/2 and 3 − 2t 2 2 2 8. (a − b)2 and a2 − b2 10. −a + 2 and −(a + 2) 12. bc − cd and c(b − d) 17 17. “Multiply by eighty percent” “Divide by eight-tenths” 18. The area of a triangle is often expressed as A = (1/2)bh. Is the expression bh/2 equivalent to the expression (1/2)bh? 19. Show that the following expressions have the same value at x = −8: 4 13.
Algebra: Form and Function by Deborah Hughes-Hallett, William G. McCallum, Eric Connally