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Algebra I Chapter 8

Algebra I Chapter 8

A polynomial that cannot

A polynomial that cannot be written as a product of two polynomials withintegral coefficients is called a prime polynomial.EXAMPLEDetermine Whether a Polynomial Is PrimeFactor 2 x 2 + 5x - 2.In this trinomial, a = 2, b = 5, and c = -2. Since b is positive, m + n ispositive. Since c is negative, mn is negative. So either m or n is negative, butnot both. Therefore, make a list of the factors of 2(-2) or -4, where onefactor in each pair is negative. Look for a pair of factors with a sum of 5.Factors of -4Sum of Factors1, -4 -3-1, 4 3-2, 2 0There are no factors with a sum of 5. Therefore, 2 x 2 + 5x - 2 cannot befactored using integers. Thus, 2 x 2 + 5x - 2 is a prime polynomial.2A. Is 4 r 2 - r + 7 prime? 2B. Is 2 x 2 + 3x - 5 prime?Solve Equations by Factoring Some equations of the form a x2+ bx + c = 0can be solved by factoring and then using the Zero Product Property.EXAMPLESolve Equations by FactoringSolve 8 a 2 - 9a - 5 = 4 - 3a. Check the solutions.8 a 2 - 9a - 5 = 4 - 3a Write the equation.8 a 2 - 6a - 9 = 0 Rewrite so that one side equals 0.(4a + 3)(2a - 3) = 0 Factor the left side.4a + 3 = 0 or 2a - 3 = 0 Zero Product Property4a = -3a = -3_4The roots are -3_4 and 3 _2 .2a = 3 Solve each equation.a = 3 _2CHECK Check each solution in the original equation.8 a 2 - 9a - 5 = 4 - 3a83_(-4) 2 - 93_(- - 5 4 - 34) (-9_2 + _ 274 - 5 4 + _ 9 4_ 254 = _ 254✔3_4)8 a 2 - 9a - 5 = 4 - 3a8 ( 3_ 2) 2 - 9 ( 3_ - 5 4 - 32) ( 3_ 2)18 - _ 272 - 5 4 - _ 9 2-_1 2 = - 1_2✔3A. 3 x 2 - 5x = 12 3B. 2 x 2 - 30x + 88 = 0Extra Examples at ca.algebra1.comLesson 8-4 Factoring Trinomials a x 2 + bx + c 443

A model for the vertical motion of a projected object is given by the equationh = -16 t 2 + vt + s, where h is the height in feet, t is the time in seconds, v isthe initial upward velocity in feet per second, and s is the initial height of theobject in feet.Factoring Whena Is NegativeWhen factoring atrinomial of the forma x 2 + bx + c where ais negative, it is helpfulto factor out a negativemonomial.PEP RALLY At a pep rally, small foamfootballs are launched by cheerleaders usinga sling-shot. How long is a football in the airif a student catches it on its way down 26 feetabove the gym floor?h = -16 t 2 + vt + sVertical motion model26 = -16 t 2 + 42t + 6 h = 26, v = 42, s = 60 = -16 t 2 + 42t - 20 Subtract 26 fromeach side.0 = -2(8 t 2 - 21t + 10) Factor out -2.0 = 8 t 2 - 21t + 10 Divide each side by -2.0 = (8t - 5)(t - 2) Factor 8 t 2 - 21t + 10.8t - 5 = 0 or t - 2 = 0 Zero Product Property8t = 5t = 2 Solve each equation.Height ofrelease6 ftt 0v 42 ft/sHeight ofreceptiont = _ 5 8The solutions are _ 5 second and 2 seconds.8The first time represents how long it takes the football to reach a height of26 feet on its way up. The later time represents how long it takes the ball toreach a height of 26 feet again on its way down. Thus, the football will be inthe air for 2 seconds before the student catches it.26 ft4. Six times the square of a number plus 11 times the number equals 2.What are possible values of x?Personal Tutor at ca.algebra1.comExamples 1–2(pp. 442–443)Example 3(p. 443)Example 4(p. 444)Factor each trinomial, if possible. If the trinomial cannot be factoredusing integers, write prime.1. 3 a 2 + 8a + 4 2. 2 t 2 - 11t + 7 3. 2 p 2 + 14p + 244. 2 x 2 + 13x + 20 5. 6 x 2 + 15x - 9 6. 4 n 2 - 4n - 35Solve each equation. Check the solutions.7. 3 x 2 + 11x + 6 = 0 8. 10 p 2 - 19p + 7 = 0 9. 6 n 2 + 7n = 2010. CLIFF DIVING Suppose a diver leaps from the edge of a cliff 80 feet abovethe ocean with an initial upward velocity of 8 feet per second. How longwill it take the diver to enter the water below?444 Chapter 8 Factoring

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