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77. Further Study of Factoring. In Chapter VI we saw how to factor certain expressions. Thus, 4 x2 — 9 ya may be factored by Formula VII into (2 x+3 y)(2 x–3 y). Likewise, such an expression as ac+bc+3 a+3 b can be factored by first writing it in the form .c(a+b)+3(a+b) and then noting that this is the same as (c+3)(a+b), all of which employs Formula III.

These are examples in which a single formula suffices to get the answer, but we often have examples in which two or more formulas are needed at the same time. Thus, in factoring a3b-ab3 we first employ Formula III to take out the factor ab. This gives a3b ab=ab(a? b). But a2 62 is itself factorable into (a+b)(a -b) by Formula VII. Therefore, the final answer here is ab(a+b)(ab). Other illustrations of this idea follow immediately below. Note that the final answer in every case contains no factors which can themselves be still further broken up into other factors.

EXAMPLE 1. Factor x2 - y2+x+y.
SOLUTION. x242+x+y=(x2y?)+(x+y)

= (x+y)(x,y)+(x+y) (Formula VII)

= (x+y)(x-y+1). Ans. (Formula III) EXAMPLE 2. Factor 4(a? 62) — 3(a+b). SOLUTION. 4(a?— 62)-3(a+b)

=4(ab)(a+b)-3(a+b) (Formula VII) =(a+b)[4(a,b)-3]

(Formula III) =(a+b)(4 a-4 6-3). Ans. EXAMPLE 3. Factor 4 q?—9 62+4 a-6 b. Solution. 4 a2—9 b2+4 a-6 b=(4 a2–962)+2(2 a-36)

= (2 a+3 b)(2 2–3 b)+2(2 a-36)

= (2 a-3 b)(2 a+3 6+2). Ans. NOTE. A common error is to think that the factoring of one or more parts of an expression is equivalent to factoring the expression itself. Thus, in the Example 3 just considered, where we had to factor 4 q2_9b2+4 a-6 b, we can of course factor the part 4 a2–962, giving (2 a+3 b) (2 a-3 b), but this does not give us the factors of the given (whole) expression, 4 22—9 b2+4 a-6 b. Observe that any expression may be said to have been factored only when it has all been put into the form of a product of two or more factors.

I.

III.

78. Summary of Factoring. All the examples in factoring which we have thus far considered have been worked by use of the following formulas : xmxn=xm+n

(§ 48) II. (xy)m=xmym.

($ 50) ab+ac=a(b+c).

($ 51) IV. x2+(m+n)x+mn=(x+m)(x+n).

(8 57) a2+2 ab+b2=(a+b)(a+b).

($ 59) VI. Q? — 2 ab+b2=(a - b)(ab).

($ 60) VII. a? — b2=(a+b)(ab).

($ 65) VIII.

(§ 71) * IX a.

a— 63 = (ab)(a? +ab+62). ($ 75) * IX 6.

q3+63=(a+b)(a? ab+62). ($ 75)

V.

хт

xn = xm-n.

EXERCISES Each expression in the following list may be factored by the formulas in § 78. Either a single one of the formulas is necessary, or several of them in the manner shown in g 77. Before attempting these, the pupil will find it desirable to review the exercises in factoring in Chapter VI and read § 77 carefully. Factor each of the following expressions.

1. x2 — ax+cx - ac.
[Hint. Write as x(x—a)+c(x—a).]
2. Yi+ya+y+1.
[Hint. Write as yề(y+1)+y+1.)
3. 28 — 22–2+1.
(Hint. Write as (23— 22)—(2–1).]

4. 2 x3 – 8 x2y+8 xyo.

[Hint. Write as 2 x(x2–4 xy+4 y2) and apply Formnla VI, $ 78.] 5. 22 – 11 x+30.

[Hint. This comes under Formula IV, § 78. See § 58.] 6. x2 +3 ax -3 a-x.

14. 1-q?— 62 2 ab. 7. a4+3 a262 — 4 64.

[Hint. Write in the form

1-(a2+62+2 ab).] 8. 83–8 XPy4.

15. a2b2c2 4 bac2. 9. 22+ y2 – 2 xy.

16. m2 +4 mn+4 n2– 16. (Hint. Rearrange the terms.] 17. 2 xy-2 22 +1. 10. a3+2 a?+4 a+8. 18. 9 a? — 6 a3ta4. 11. 24 – 13 x2+36.

19. a’na+a_m2 62m2 ban?. 12: 24+y4 – 2 x?y?.

20. ax b-a+bx. 13. 24—(x - 2)2

21. ab2 2 abc+ac. Hint. The answer contains 22. 1+9c4 +6 c. three factors.]

23. (x2— 1)2+(2x+3)(x-1)2. 24. a? — 64 - a2x2 +64x2. 25. 3 x3 – 3 x+4 x4—4 x. 26. a4—4 b4+q2+2 62. 27. 1-4 aoboco -9 couz+12 abcxyz. 28. xy-1+x-y. 29. a2+(62 bx2)ay2 62x2y2. 30. 1-a2b2 — x?y2 +2 abxy.

31. (a+b)2(x - y)(a+b)(x2 - y2). 32. (22 - y2)2 – (x2 — xy)2. 38. (1–2 x)2 — 24. 33. m3+n2 mn-mn?. 39: wsy–10 x2y2z2+25 xy3z4. *34. 23+y3+x2 — yề.

40. x4 – 18 x2 +81. *35. (x+1)3— 26.

*41. 1+(x+1)3. *36. 5 x?y-5 xy*.

*42. x3+15 x2+75 x+125. *37. 23 – 27 7(x-3). *43. 3 ab(a+b)+23+63.

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LAPLACE

(Pierre Simon Laplace, 1749–1827) Famous in mathematics for his researches, which were of a most advanced kind, and especially famous in astronomy for his enunciation of the Nebular Hypothesis. Interested also in physics and at various times held high political offices under Napoleon.

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