Thursday, March 24, 2011

Math problems of the week: factoring polynomials in the 21st century vs. the 1900's

I. From the Chapter Review of the "Factoring" chapter of the University of Chicago School Mathematics Project Algebra text (second edition), (Chapter 12, pp. 769-771):

SPUR stands for Skills, Properties, Uses, and Representations. The Chapter Review questions are grouped according to the SPUR objectives for this chapter.

SKILLS DEAL WITH PROCEDURES USED TO GET ANSWERS

Objective C: Factor quadratic expressions

18. d2 - 8d - 20
...
20. 11a2 - 26a + 21 =
(a) (11a - 7)(a - 3)
(b) (11a + 7)(a - 3)
(c) (11a - 7)(a + 3)
(c) (11a + 7)(a + 3)
...
24. -3 - 2k + 8k2
...
Write the difference of squares as the product of two binomials:
30. 25t2 - 25

Objective D: Solve quadratic equations by factoring
31. x2 - 2x = 0
...
40. 0 = 16m2 - 8m + 1

PROPERTIES DEAL WITH THE PRINCIPLES BEHIND THE MATHEMATICS

Objective G: Determine whether a quadratic polynomial can be factored over the integers.

53. Multiple choice. Which polynomial can be factored over the integers?
(a) x2 - 11
(b) x2 + 11
(c) x2 - 121
(d) x2 + 112

54. Suppose m, n, and p are integers. When wil the quadratic expression mx2 + nx + p be factorable over the integers?

Use the discriminant to determine whether the expression can be factored over the integers.
...
57. 3r2 + 2r - 21.

58. In attempting to factor x2 - 16x + 20, Rachel made a list of pairs of factors of 20 and checked the sum of each pair.

Factors of 20:
-1, -20
-2, -10
-4, -5
Sums of factors:
-21
-12
-9

From this list she deduced that x2 - 16x + 20 was not factorable over the integers. Determine whether she was right or wrong. Explain your answer.

59. Find two integer factors whose sum in 10. What does this tell you about x2 + 10x + 24?

USES DEAL WITH APPLICATIONS OF MATHEMATICS IN REAL SITUATIONS

When a old ball is hit with an upward velocity of 80 feet per second, an equation that gives its height (in feet) above the ground after t seconds is h = 80t - 16t2.
72. How long with the golf ball be in the air?

REPRESENTATIONS DEAL WITH PICTURES, GRAPHS, OR OBJECTS THAT ILLUSTRATE CONCEPTS

Objective J: Represent quadratic expressions and their factorizations with areas.

78. Show that x2 + 7x + 6 can be factored by arranging tiles representing the polynomial in a rectangle. Sketch your argument.

79. A square has an area of 9a2 + 30ab + 25b2. What is the length of a side of the square?

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II. From the chapter review of the "Factors" chapter of Wentworth's New School Algebra (published in 898) (Chapter 7, pp. 105-107):

1. a2 - 9a
...
11. a2 - (m + n)2
...
21. x2 - 7x + 6
...
31. (x - y)2 - b2
...
41. 3x4 - 6x3 + 9x2
...
51. 16x4 - 81
...
61. a3 - b3 + a - b
...
71. a2 + a + 3b - 9b2
...
81. a2 - 2ab + b2 + 12xy - 4x2 - 9y2
...
91. x4 + 8x2 - 9
...
101. x3 - y3 - 3xy(x - y)
...
111. 25a2 - 4x2 + 4x - 10a
...
121. x4 - 2abx2 - a4 - a2b2 - b4

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III. Extra Credit

Which problem set presents a greater variety of factoring problems?

Which is more likely: that someone who can complete the first set of factoring problems can also complete the second set of factoring problems, or vice versa?

4 comments:

Eowyn said...

Question 20 has no correct answer, by the way.

Just for bonus fun.

FedUpMom said...

I was wondering about that too. I think Katharine must have mistakenly cobbled together two different problems.

Notice that the question has x as the variable, but the putative "solutions" have a as the variable.

Katharine Beals said...

Yes, two problems mistakenly cobbled together; now fixed. Thanks for noticing, guys!

Anonymous said...

You should take a look at the method CPM uses to factor polynomials in their Algebra Connections book. A student "schooled" using the CPM approach could probably factor problems 1 and 21 from the Wentworth book...and that's it.