Algebra I : Semester II : Polynomials

Sections:

Introduction  |   Section 1  |   Section 2  |   Section 3  |  Section 4  |  Section 5  |  Section 6

Section Four

Part 1  |  Part 2  |  Part 3  |  Part 4

Algebra 1: Section 4: Polynomials Solving Equations Using Factoring   Click here for a presentation on Solving Equations with Factoring. Now, let’s write the rule and techniques learned in the presentation.   Zero Product Property If the product of two quantities is zero, then one of the quantities must equal zero. If ab = 0, then a = 0 or b = 0. Use this property to solve equations in factored form that are equal to zero.   Steps for Solving Equations using Factoring Set the equation equal to zero. Factor the polynomial if it is not already in factored form. Set each factor equal to zero. Solve each of the equations and write the solution set. Check. Example 1:  Solving In Factored Form Solve and check the solution:   (x + 5) (4x – 5) = 0. (x + 5) (4x – 5) = 0     It is already in factored form x + 5 = 0    or x = 0 – 5 x = -5 4x – 5 = 0 4x = 0 + 5 4x = 5 x = 5/4 x = 1 ¼ or 1.25 Check. (x + 5) (4x – 5) = 0 (-5 + 5) (4 ∙ -5 – 5) = 0 (0) (-25) = 0 0 = 0  √ (x + 5) (4x – 5) = 0 (1.25 + 5) (4 ∙ 1.25 – 5) = 0 (6.25) (0) = 0 0 = 0   √ Therefore the solution set is {-5, 1.25}. Note: Do not use a decimal if you would have a repeating or non-terminating decimal solution. Use a fraction in those cases for the exact solution. Example 2:  Factoring to Solve Solve and check the solution:  2x2 + 162x = 0. 2x2 + 162x = 0 2x (x + 81) = 0      Factors 2x = 0      or x = 0/2 x = 0 x + 81 = 0 x = 0 – 81 x = -81 Check. 2x2 + 162x = 0 2(0)2 + 162(0) = 0 0 + 0 = 0 0 = 0  √ 2x2 + 162x = 0 2(-81)2 –+162(-81) = 0 2(6561) + 162(-81) = 0 13,122 + - 13,122 = 0 0 = 0  √ The solution is {0, -81}. Example 3:  Writing the Equation Equal to Zero First Solve 8x2 = 6x. Write this to equal zero in order to use the technique; subtract 6x on both sides of the equation. 8x2 = 6x 8x2 – 6x = 6x – 6x 8x2 – 6x = 0 2x (4x – 3 ) = 0  Factors 2x = 0      or x = 0/2 x = 0 4x – 3 = 0 4x = 0 + 3 4x = 3 x = ¾ or 0.75 Check. 8x2 = 6x 8(0)2 = 6(0) 8(0) = 0 0 = 0  √ 8x2 = 6x 8(0.75)2 = 6(0.75) 8( 0.5625) = 4.5 4.5 = 4.5  √ Therefore, the solution is {0, 0.75} Example 4:  Problem Solving If an automatic launcher, sitting on the ground, launches a rocket at 20 feet per second, how many seconds will it take to fall back to the ground? Use the equation h = -16t2 + 20t, where h is the height in feet and t is the number of seconds. You know that the height of an object sitting on the ground is zero feet. So, let h equal zero. h = -16t2 + 20t 0 = -16t2 + 20t 0 = 4t (-4t + 5)         Factor 4t = 0        or t = 0/4 t = 0 At 0 seconds we are just starting the rocket. -4t + 5 = 0 -4t = 0 – 5 -4t = -5 t = -5/-4 t = 1.25 At 1.25 seconds the rocket should land on the ground. Check (you only need to check for t = 1.25). 0 = -16t2 + 20t 0 = -16(1.25)2 + 20(1.25) 0 = -16(1.5625) + 20(1.25) 0 = -25 + 25 0 = 0  √ Since zero seconds is the moment when the rocket was launched, it took 1.25 seconds for the rocket to fall to the ground   Practice Solve each of the following and check your solution. Equation Factored Form Solution x (x – 14) = 0 x (x – 14) = 0 {0, 14} b (b  + 19) = 0 b (b  + 19) = 0 {0, -19} (m + 4) (3m – 15) = 0 (m + 4) (3m – 15) = 0 {-4, 5} (2x + 3) (3x – 8) = 0 (2x + 3) (3x – 8) = 0 {-3/2, 8/3}  (4w + 5) (6w  + 12) = 0 (4w + 5) (6w  + 12) = 0 {-5/4, -2} or {-1.25, -2} 5z2 + 20z = 0 5z (z + 4) = 0 {0, -4} 2x2 – 5x = 0 x (2x – 5) = 0 {0, 5/2} or {0, 2.5} 12x2 = 4x Hint: Subtract 4x first to set = 0. 12x2 – 4x = 0 4x (3x – 1) = 0 {0, 1/3} 20x2 = -15x 20x2 + 15x = 0 5x (4x + 3) = 0 {0, -3/4} or {0, -0.75} Problem Solving Annie hit a softball at an initial velocity of 35 feet per second (ft/sec). It was caught four feet above the ground. How many seconds was the ball in the air? Use the equation 4 = -16t2 + 35t + 4. 4 – 4 = -16t2 + 35t + 4 – 4             Subtract 4 to set the equation to zero. 0 = -16t2 + 35t                                Simplify 0 = t (-16t + 35)                              Factor -16t + 35 = 0       or     t = 0            Set = 0 -16t = 0 – 35                                  Solve -16t = -35 t = -35/-16 t = 2.1875 {0, 2.1875} Since zero seconds is the moment when the ball was hit, it took approximately 2.2 seconds for the ball to be caught. Section 4 Homework(10 points) It’s time to complete your homework. Find the section homework link to submit it for a grade. You may take this only one time, so check your understanding of the material before completing your homework, and do your best. Now go on to the next section

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