Solving Polynomial Equations by Factoring Quick Review FACTORING

Solving Polynomial Equations by Factoring

Quick Review � FACTORING a polynomial means to break it apart into its prime factors. � For example: �x 2 – 4 = (x + 2)(x – 2) �x 2 + 6 x + 5 = (x + 1)(x + 5) � 3 y 2 + 10 y – 8 = (3 y – 2)(y + 4)

In this lesson, we want to factor polynomials that have four terms. To do this, we will use a method called grouping.

For example, let’s factor ax + ay + bx + by. � Let’s begin by grouping the first two terms and the last two terms: �(ax + ay) + (bx + by) � Next, factor out the GCF of each group: �a(x + y) + b(x + y) � Notice that (x + y) is a common factor of each term. This means: ◦ ax + ay + bx + by = (x + y)(a + b)

Factor: x 3 + 7 x 2 + 2 x + 14 � First, group each set of terms: �(x 3 + 7 x 2) + (2 x + 14) � Then, take out GCFs: �x 2(x + 7) + 2(x + 7) � Write final answer: �(x + 7)(x 2 + 2)

Factor: 3 x 2 + xy – 12 x – 4 y � First, group each set of terms: �(3 x 2 + xy) – (12 x + 4 y) �Notice that because there was a minus here, we needed to change the sign in our second group! � Then, take out GCFs: �x(3 x + y) – 4(3 x + y) � Write final answer: �(3 x + y)(x – 4)

You try these. When you are ready to check your answers, move to the next slide. 1. a 3 – 2 a 2 + 5 a – 10 2. 15 y 3 + 24 y 2 – 35 y – 56

How well did you do? 1. a 3 – 2 a 2 + 5 a – 10 2. 15 y 3 + 24 y 2 – 35 y – 56 Group: (a 3 – 2 a 2) + (5 a – 10) Group: (15 y 3 + 24 y 2) – (35 y + 56) **Notice the sign change! GCFs: a 2(a – 2) + 5(a – 2) GCFs: 3 y 2(5 y + 8) – 7(5 y + 8) Answer: (a – 2)(a 2 + 5) Answer: (5 y + 8)(3 y 2 – 7)

Earlier in the course, you learned how to solve polynomial equations by factoring.

For example, let’s solve the equation 3 x 2 + 2 x = 5. � First, remember that your equation must be set equal to 0 before factoring. � 3 x 2 + 2 x – 5 = 0 � There isn’t a GCF to factor out. So, factor. �(3 x + 5)(x – 1) = 0 � Next, using the Zero Product Property, set each factor equal to 0 and solve for x. � 3 x + 5 = 0 or x – 1 = 0 �x = -5/3 or x = 1 � The solution set is: Note the graph shows us 2 zeros, so our solution makes sense!

Now, let’s solve some more challenging polynomial equations!

Solve: 4 x 5 – 6 x 4 – 4 x 3 = 0 � These three terms have a GCF that we will need to factor out first: � 2 x 3(2 x 2 – 3 x – 2) = 0 � Then, factor: � 2 x 3(2 x + 1)(x – 2) = 0 � Set each factor equal to 0: � 2 x 3 = 0 or 2 x + 1 = 0 or x – 2 = 0 � Solve for x: Note the graph shows us 3 zeros, so our solution makes sense!

Solve: 3 x 4 – 4 x 2 = -1 � First, we need to set our equation equal to 0: � 3 x 4 – 4 x 2 + 1 = 0 � Then, factor: �(3 x 2 – 1)(x 2 – 1) = 0 � Set each factor equal to 0: � 3 x 2 – 1 = 0 or x 2 – 1 = 0 � Solve for x: Note the graph shows us 4 zeros, so our solution makes sense!

Solve: x 4 – 1 = 0 � First, factor: �(x 2 + 1)(x 2 – 1) = 0 �(x 2 + 1)(x – 1) = 0 � Set each factor equal to 0: �x 2 + 1 = 0 or x – 1 = 0 � Solve for x: Note the graph shows us 2 zeros, because our calculator only shows REAL zeros.

One of the most widely used applications of solving polynomial equations is modeling expenses and profit. Often times, the sales of a particular item can be modeled using a polynomial equation. Companies can use these equations to predict what prices will result in the largest profit margins.

For example: �A company sells an item for x dollars. Their revenue y (in dollars) is given by the polynomial equation y = -10 x 4 + 1000 x 2. At what price will the company stop making money?

Example continued � Look at the graph of the function first: � We want to find the zeros, because then we will know when the company isn’t making any money (revenue of $0). � We can factor the polynomial equation to find these zeros!!!

Example continued � Let’s factor 0 = -10 x 2(x 2 – 100) � Now set each factor equal to 0 and solve for x: � 0 = -10 x 2(x + 10)(x – 10) �-10 x 2 = 0 or x + 10 = 0 or x – 10 = 0 � -10 doesn’t make sense in the situation because the company wouldn’t charge -$10 � 0 doesn’t make sense in the situation because the company wouldn’t charge $0 � 10 does make sense! If the company charges $10, they won’t make any money…this would most likely be because people wouldn’t spend that much money for the item.