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2 6.2 Factoring Trinomials

3 What You Will Learn  Factor trinomials of the form x 2 + bx + c  Factoring trinomials in two variables  Factor trinomials completely

4 Factoring Trinomials of the Form x 2 + bx + c

5 You know that the product of two binomials is often a trinomial. Here are some examples. Factored Form F O I L Trinomial Form (x – 1)(x + 5) = x 2 + 5x – x – 5 = x 2 + 4x – 5 (x – 3)(x – 3) = x 2 – 3x – 3x + 9 = x 2 – 6x + 9 (x + 5)(x + 1) = x 2 + x + 5x + 5 = x 2 + 6x + 5 (x – 2)(x – 4) = x 2 – 4x – 2x + 8 = x 2 – 6x + 8 Try covering the factored forms in the left-hand column above.

6 Factoring Trinomials of the Form x 2 + bx + c Can you determine the factored forms from the trinomial forms? In this section, you will learn how to factor trinomials of the form x 2 + bx + c. To begin, consider the following factorization.

7 Factoring Trinomials of the Form x 2 + bx + c So, to factor a trinomial x 2 + bx + c into a product of two binomials, you must find two numbers m and n whose product is c and whose sum is b. There are many different techniques that can be used to factor trinomials. The most common technique is to use guess, check, and revise with mental math. For example, try factoring the trinomial x 2 + 5x + 6.

8 Factoring Trinomials of the Form x 2 + bx + c You need to find two numbers whose product is 6 and whose sum is 5. Using mental math, you can determine that the numbers are 2 and 3.

9 Example 1 – Finding the Greatest Common Factor Factor the trinomial x 2 + 5x – 6. Solution: You need to find two numbers whose product is –6 and whose sum is 5.

10 Example 2 – Finding the Greatest Common Factor Factor the trinomial x 2 – x – 6. Solution:

11 Factoring Trinomials of the Form x 2 + bx + c If you have trouble factoring a trinomial, it helps to make a list of all the distinct pairs of factors and then check each sum. For instance, consider the trinomial x 2 – 5x – 24 = (x + )(x – ). In this trinomial the constant term is negative, so you need to find two numbers with opposite signs whose product is –24 and whose sum is –5. Opposite signs

12 Factoring Trinomials of the Form x 2 + bx + c Factors of –24 Sum 1, –24 –23 –1, , –12 –10 –2, , –8 –5 –3, 8 5 4, –6 –2 –4, 6 2 So, x 2 – 5x – 24 = (x + 3)(x – 8). Correct choice

13 Factoring Trinomials of the Form x 2 + bx + c With experience, you will be able to narrow the list of possible factors mentally to only two or three possibilities whose sums can then be tested to determine the correct factorization. Here are some suggestions for narrowing the list.

14 Factoring Trinomials in Two Variables

15 Factoring Trinomials in Two Variables The next example show how to factor trinomials of the form x 2 + bxy + cy 2. Note that this trinomial has two variables, x and y. However, from the factorization x 2 + bxy + cy 2 = x 2 + (m + n)xy + mny 2 = (x + my)(x + ny) you can see that you still need to find two factors of c whose sum is b.

16 Example 5 – Factoring a Trinomial in Two Variables Factor the trinomial x 2 – xy – 12y 2. Solution: You need to find two numbers whose product is –12 and whose sum is –1.

17 Example 6 – Factoring a Trinomial in Two Variables Factor the trinomial x xy + 10y 2. Solution: You need to find two numbers whose product is 10and whose sum is 11.

18 Example 7 – Factoring a Trinomial in Two Variables Factor the trinomial y 2 – 6xy + 8x 2. Solution: You need to find two numbers whose product is 8 and whose sum is –6.

19 Factoring Completely

20 Factoring by Grouping Some trinomials have a common monomial factor. In such cases you should first factor out the common monomial factor. Then you can try to factor the resulting trinomial by the methods of this section. This “multiple-stage factoring process” is called factoring completely. The trinomial below is completely factored. 2x 2 – 4x – 6 = 2(x 2 – 2x – 3) = 2(x – 3)(x + 1) Factor out common monomial factor 2. Factor trinomial.

21 Example 8 – Factoring Completely Factor the trinomial 2x 2 – 12x + 10 completely. Solution: 2x 2 – 12x + 10 = 2(x 2 – 6x + 5) = 2(x – 5)(x – 1) Factor out common monomial factor 2. Factor trinomial.

22 Example 9 – Factoring Completely Factor the trinomial 3x 3 – 27x x completely. Solution: 3x 3 – 27x x = 3x(x 2 – 9x + 18) = 3x(x – 3)(x – 6) Factor out common monomial factor 3x. Factor trinomial.

23 Example 10 – Factoring Completely Factor the trinomial 47y 4 – 32y y 2 completely. Solution: 4y 4 – 32y y 2 = 4y 2 (y 2 + 8y + 7) = 4y 2 (y + 1)(y + 1) Factor out common monomial factor 4y 2. Factor trinomial.

24 Example 11 – Geometry: Volume of an Open Box An open box is to be made from a four-foot-by-six-foot sheet of metal but cutting equal squares from the corners and turning up the sides. The volume of the box can be modeled by V = 4x 3 – 20x x, 0 < x < 2. a.Factor the trinomial that models the volume of the box. Use the factored form to explain how the model was found. b.Use a spreadsheet to approximate the size of the squares to be cut from the corners so that the box has the maximum volume.

25 cont’d Solution a.4x 3 – 20x x= 4x(x 2 – 5x + 6) = 4x(x – 3)(x – 2) Because 4 = (–2)(–2), you can rewrite the factored form as 4x(x – 3)(x – 2)= x[(–2)(x – 3)][(–2)(x – 2)] = x(6 – 2x)(4 – 2x) = (6 – 2x)(4 – 2x)(x) The model was found by multiplying the length, width, and height of the box. Factor out common monomial factor 4x Example 11 – Geometry: Volume of an Open Box Factored form

26 cont’d b. From the spreadsheet below, you can see the maximum volume of the box is about 8.45 cubic feet. This occurs when the value of x is about 0.8 feet. Example 11 – Geometry: Volume of an Open Box