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204 Chapter 8 8–1 ◆ Factors and Factoring Common Factors Factors of an Expression The factors of an expression are those quantities whose product is the original expression. ◆◆◆ Example 1: The factors of x2 9 are x 3 and x 3, because (x 3)(x 3) x2 3x 3x 9 x2 9 ◆◆◆ Prime Factors Many expressions have no factors other than 1 and themselves. They are called prime expressions. ◆◆◆ Example 2: Two factors of 30 are 5 and 6. The 5 is prime because it cannot be factored. However, the 6 is not prime because it can be factored into 2 and 3. Thus the prime factors of ◆◆◆ 30 are 2, 3, and 5. Example 3: The expressions x, x 3, and x2 3x 7 are all prime. They cannot be fac◆◆◆ tored further. ◆◆◆ Example 4: The expressions x2 9 and x2 5x 6 are not prime because they can be ◆◆◆ factored. They are said to be factorable. ◆◆◆ Factoring Factoring is the process of finding the factors of an expression. It is the reverse of finding the product of two or more quantities. Multiplication (Finding the Product) x (x 4) x2 4x Factoring (Finding the Factors) x2 4x x (x 4) We factor an expression by recognizing the form of that expression and then applying standard rules for factoring. In the first type of factoring we will cover, we look for common factors. Common Factors If each term of an expression contains the same quantity (called the common factor), that quantity may be factored out. This is simply the distributive law, which we studied earlier. Common Factor ◆◆◆ ab ac a(b c) 5 Example 5: In the expression 2x x2 each term contains an x as a common factor. So we write 2x x2 x (2 x) Most of the factoring we will do will be of this type. ◆◆◆ Section 8–1 ◆◆◆ (a) (b) (c) (d) ◆ 205 Common Factors Example 6: 3x3 2x 5x4 x(3x2 2 5x3) 3xy2 9x3y 6x2y2 3xy(y 3x2 2xy) 3x3 6x2y 9x4y2 3x2(x 2y 3x2y2) 2x2 x x (2x 1) ◆◆◆ When factoring, don’t confuse superscripts (powers) with subscripts. ◆◆◆ Example 7: The expression x3 x2 is factorable. x3 x2 x2(x 1) But the expression x 3 x2 ◆◆◆ is not factorable. Common Error Students are sometimes puzzled over the “1” in Example 7. Why should it be there? After all, when you remove a chair from a room, it is gone; there is nothing (zero) remaining where the chair used to be. If you remove an x by factoring, you might assume that nothing (zero) remains where the x used to be. 2x2 x x (2x 0)? Prove to yourself that this is not correct by multiplying the factors to see if you get back the original expression. Factors in the Denominator Common factors may appear in the denominators of the terms as well as in the numerators. ◆◆◆ Example 8: 1 2 1p 2q (a) 1 x x2 x x 2 x x 2x x 1 2 (b) p x q y 3y y y 3 y2 ◆◆◆ Checking To check if factoring has been done correctly, simply multiply your factors together and see if you get back the original expression. ◆◆◆ Example 9: Are 2xy and x 3 y2 the factors of 2x2y 6xy 2xy3? Solution: Multiplying the factors, we obtain 2xy(x 3 y2) 2x2y 6xy 2xy3 (checks) ◆◆◆ This check will tell us whether we have factored correctly but, of course, not whether we have factored completely (that is, found the prime factors). 206 Chapter 8 ◆ Factors and Factoring CASE STUDY – CHECKING YOUR FACTORING Here is an interesting look at mathematics: Although it may seem a little strange to have x y, there is actually nothing wrong with it. Sometimes you will see an x changed to a y or a y changed to an x. Normally, you can’t do that unless x y, as is the case here. Look over the steps below and notice the factoring. Use the knowledge you gain in this chapter to check whether the factoring is done correctly. If everything is correct, then two is the same as one and all of our mathematics developed over the last several thousand years is wrong. Can you determine what is wrong in this chain of math? x2 y 2 x2 xy (x y)(x y) x(x y) (x y)(x y) x(x y) (x y) (x y) xyx given x y xxx 2x x 21 Exercise 1 ◆ Common Factors Factor each expression and check your results. 1. 2. 3. 4. 3y2 y3 6x 3y x5 2x4 3x3 9y 27xy 5. 3a a2 3a3 6. 8xy3 6x2y2 2x3y 7. 5(x y) 15(x y)2 a a2 a3 8. 3 4 5 3 5 2 9. x x2 x3 3ab2 6a2b 12ab 10. y y3 y2 5m 15m2 25m3 11. 2n 8n 4n2 16y2 8y3 8y4 12. 3x 9x2 3x3 13. 5a2b 6a2c 14. 15. 16. 17. 18. 19. a2c b2c c2d 4x2y cxy2 3xy3 4abx 6a2x2 8ax 3a3y 6a2y2 9ay3 2a2c 2a2c2 3ac 5acd 2c2d 2 bcd 20. 4b2c2 12abc 9c2 21. 8x2y2 12x2z2 Section 8–2 ◆ 207 Difference of Two Squares 22. 6xyz 12x2y2z 23. 3a2b abc abd 24. 5a3x2 5a2x3 10a2x2z Applications 25. When a bar of length L0 is changed in temperature by an amount t, its new length L will be L L0 L0 t, where is the coefficient of thermal expansion. Factor the right side of this equation. 26. A sum of money a when invested for t years at an interest rate n will accumulate to an amount y, where y a ant. Factor the right side of this equation. 27. When a resistance R1 is heated from a temperature t1 to a new temperature t, it will increase in resistance by an amount (t t1)R1, where is the temperature coefficient of resistance. The final resistance will then be R R1 (t t1)R1. Factor the right side of this equation. 28. An item costing P dollars is reduced in price by 15%. The resulting price C is then C P 0.15P. Factor the right side of this equation. 29. The displacement of a uniformly accelerated body is given by a s v0t t2 2 Factor the right side of this equation. 30. The sum of the voltage drops across the resistors in Fig. 8–1 must equal the battery voltage E. Compare your result for problem 27 with Eq. A70. E R1 i E iR1 iR2 iR3 Factor the right side of this equation. 31. The mass of a spherical shell having an outside radius of r2 and an inside radius r1 is 4 4 mass r23D r13D 3 3 where D is the mass density of the material. Factor the right side of this equation. 8–2 R3 FIGURE 8–1 Difference of Two Squares Form As we saw in Sec. 2–4, an expression of the form a2 b2 perfect square perfect square minus sign where one perfect square is subtracted from another, is called a difference of two squares. It arises when (a b) and (a b) are multiplied together. Difference of Two Squares a2 b2 (a b)(a b) Factoring the Difference of Two Squares Once we recognize its form, the difference of two squares is easily factored. 41 R2