0.4 FACTORING POLYNOMIALS


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1 36_.qxd /3/5 :9 AM Page 9 SECTION. Factoring Polynomials 9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use synthetic division to factor polynomials of degree three or more. Use the Rational Zero Theorem to find the real zeros of polynomials. Factorization Techniques The Fundamental Theorem of Algebra states that every nthdegree polynomial a n x n a n x n... a x a, a n has precisely n zeros. (The zeros may be repeated or imaginary.) The problem of finding the zeros of a polynomial is equivalent to the problem of factoring the polynomial into linear factors. Special Products and Factorization Techniques Quadratic Formula ax bx c Special Products x a x ax a x 3 a 3 x ax ax a x 3 a 3 x ax ax a x a x ax ax a x b ± b ac a Example x 3x Examples x x 9 x 3x 3 x 3 8 x x x x 3 6 x x x 6 x 6 x x x 3 ± 3 Binomial Theorem Factoring by Grouping acx 3 adx bcx bd ax cx d bcx d ax bcx d Examples x a x ax a x 3 x 6x 9 x a x ax a x a 3 x 3 3ax 3a x a 3 x 5 x x 5 x a 3 x 3 3ax 3a x a 3 x 3 x 3 6x x 8 x a x ax 3 6a x a 3 x a x 3 x 3 3x 3x x x 8x 3 x 3x 6 x a x ax 3 6a x a 3 x a x x 6x 3 96x 56x 56 nn nn n x a n x n nax n a *! x n a 3! 3 x n3... na n x a n nn nn n x a n x n nax n a! x n a 3! 3 x n3... ± na n x a n Example 3x 3 x 6x x 3x 3x x 3x * The factorial symbol! is defined as follows:!,!,!, 3! 3 6,! 3, and so on.
2 36_.qxd /3/5 :9 AM Page   CHAPTER A Precalculus Review EXAMPLE Applying the Quadratic Formula Use the Quadratic Formula to find all real zeros of each polynomial. (a) x 6x (b) x 6x 9 (c) x 6x 5 SOLUTION STUDY TIP Try solving Example (b) by factoring. Do you obtain the same answer? (a) Using a, b 6, and c, you can write x b ± b ac a So, there are two real zeros: x 3 5 (b) In this case, a, b 6, and c 9, and the Quadratic Formula yields x b ± b ac a and So, there is one (repeated) real zero: x 3. (c) For this quadratic equation, a, b 6, and c 5. So, x b ± b ac a Because is imaginary, there are no real zeros ± ± 8 6 ± ± 5 3 ± 5. x 6 ± ± ±..9. TRY IT Use the Quadratic Formula to find all real zeros of each polynomial. (a) x x (b) x 8x 6 (c) x x 5 The zeros in Example (a) are irrational, and the zeros in Example (c) are imaginary. In both of these cases the quadratic is said to be irreducible because it cannot be factored into linear factors with rational coefficients. The next example shows how to find the zeros associated with reducible quadratics. In this example, factoring is used to find the zeros of each quadratic. Try using the Quadratic Formula to obtain the same zeros.
3 36_.qxd /3/5 :9 AM Page  SECTION. Factoring Polynomials  EXAMPLE Factoring Quadratics Find the zeros of each quadratic polynomial. (a) x 5x 6 (b) x 6x 9 (c) x 5x 3 SOLUTION (a) Because x 5x 6 x x 3 the zeros are x and x 3. (b) Because x 6x 9 x 3 the only zero is x 3. (c) Because x 5x 3 x x 3 the zeros are x and x 3. ALGEBRA REVIEW The zeros of a polynomial in x are the values of x that make the polynomial zero. To find the zeros, factor the polynomial into linear factors and set each factor equal to zero. For instance, the zeros of x x 3 occur when x and x 3. TRY IT Find the zeros of each quadratic polynomial. (a) x x 5 (b) x x (c) x 7x 6 EXAMPLE 3 Finding the Domain of a Radical Expression Find the domain of x 3x. SOLUTION Because x 3x x x you know that the zeros of the quadratic are x and x. So, you need to test the sign of the quadratic in the three intervals,,,, and,, as shown in Figure.5. After testing each of these intervals, you can see that the quadratic is negative in the center interval and positive in the outer two intervals. Moreover, because the quadratic is zero when x and x, you can conclude that the domain of x 3x is,,. Domain Values of x 3x x x 3x.5 Undefined 3 x 3x + x 3x + x 3x + is defined. is not defined. is defined. 3 FIGURE.5 x TRY IT 3 Find the domain of x x.
4 36_.qxd /3/5 :9 AM Page   CHAPTER A Precalculus Review Factoring Polynomials of Degree Three or More It can be difficult to find the zeros of polynomials of degree three or more. However, if one of the zeros of a polynomial is known, then you can use that zero to reduce the degree of the polynomial. For example, if you know that x is a zero of x 3 x 5x, then you know that x is a factor, and you can use long division to factor the polynomial as shown. x 3 x 5x x x x x x x As an alternative to long division, many people prefer to use synthetic division to reduce the degree of a polynomial. Synthetic Division for a Cubic Polynomial Given: x x x is a zero of ax 3 bx cx d. a b c d Vertical pattern: Add terms. a Coefficients for quadratic factor Diagonal pattern: Multiply by x. Performing synthetic division on the polynomial x 3 x 5x using the given zero, x, produces the following. When you use synthetic division, remember to take all coefficients into account even if some of them are zero. For instance, if you know that x is a zero of x 3 3x 5 x x x x 3 x 5x you can apply synthetic division as shown. 3 7 x x x 7 x 3 3x STUDY TIP Note that synthetic division works only for divisors of the form x x. Remember that x x x x. You cannot use synthetic division to divide a polynomial by a quadratic such as x 3.
5 36_.qxd /3/5 :9 AM Page 3 The Rational Zero Theorem There is a systematic way to find the rational zeros of a polynomial. You can use the Rational Zero Theorem (also called the Rational Root Theorem). SECTION. Factoring Polynomials 3 Rational Zero Theorem If a polynomial a n x n a n x n... a x a has integer coefficients, then every rational zero is of the form where p is a factor of a, and q is a factor of a n. x pq, EXAMPLE Using the Rational Zero Theorem Find all real zeros of the polynomial. x 3 3x 8x 3 SOLUTION Factors of constant term: ±, ±3 Factors of leading coefficient: ±, ± The possible rational zeros are the factors of the constant term divided by the factors of the leading coefficient.,, 3, 3,,, 3, 3 x 3 3x 8x 3 By testing these possible zeros, you can see that x works Now, by synthetic division you have the following x x 5x 3 x 3 3x 8x 3 Finally, by factoring the quadratic, x 5x 3 x x 3, you have x 3 3x 8x 3 x x x 3 and you can conclude that the zeros are x, x, and x 3. STUDY TIP In Example, you can check that the zeros are correct by substituting into the original polynomial. Check that x is a zero Check that x is a zero Check that x 3 is a zero TRY IT Find all real zeros of the polynomial. x 3 3x 3x
6 36_.qxd /3/5 :9 AM Page   CHAPTER A Precalculus Review EXERCISES. In Exercises 8, use the Quadratic Formula to find all real zeros of the seconddegree polynomial.. 6x x. 8x x 3. x x 9. 9x x 5. y y 6. x 6x 7. x 3x 8. 3x 8x In Exercises 9 8, write the seconddegree polynomial as the product of two linear factors. 9. x x. x x 5. x x. 9x x 3. x x. x x 5. 3x 5x 6. x xy y 7. x xy y 8. a b abc c In Exercises 9 3, completely factor the polynomial y. x 6. x 3 8. y y 3 6. z x x a 3 b 3 7. x 3 x x 8. x 3 x x 9. x 3 3x x 6 3. x 3 5x 5x 5 3. x 3 x x 3. x 3 7x x x 5x 6 3. x 9x 5 In Exercises 35 5, find all real zeros of the polynomial. 35. x 5x 36. x 3x 37. x x x 3. x 8. x 3 9. x 8 3. x x. x 5x 6 5. x 5x 6 6. x x 7. x x x 6 5. x x 3 x x 5. x 3 x 6x 3 In Exercises 53 56, find the interval (or intervals) on which the given expression is defined. 53. x 5. x 55. x 7x 56. x 8x 5 In Exercises 57 6, use synthetic division to complete the indicated factorization. 57. x 3 3x 6x x 58. x 3 x x x 59. x 3 x x x 6. x 6x 3 96x 56x 56 x In Exercises 6 68, use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. 6. x 3 x x 8 6. x 3 7x x 3 6x x 6 6. x 3 x 5x x 3 x 9x x 3 9x 8x 67. x 3 3x 3x 68. x 3 x 3x Average Cost The minimum average cost of producing x units of a product occurs when the production level is set at the (positive) solution of.3x. Determine this production level. 7. Profit The profit P from sales is given by P x x 38 where x is the number of units sold per day (in hundreds). Determine the interval for x such that the profit will be greater than. 7. Chemistry: Finding Concentrations Use the Quadratic Formula to solve the expression.8 5 which is needed to determine the quantity of hydrogen ions H in a solution of. M acetic acid. Because x represents a concentration of H, only positive values of x are possible solutions. (Source: Adapted from Zumdahl, Chemistry, Sixth Edition) 7. Finance After years, an investment of $ is made at an interest rate r, compounded annually, that will yield an amount of A r. x. x Determine the interest rate if A $3.
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