Example 1 Factor ax2 + bx + c Example 2 Factor When a, b, and c Have a Common Factor Example 3 Determine Whether a Polynomial Is Prime Example 4 Solve Equations by Factoring Lesson 4 Contents

Example 1 Factor ax2 + bx + c

Step 1: Draw a box with 9 squares Factor In order to factor a trinomial with an “a” value we will use a technique called the factor box. Step 1: Draw a box with 9 squares Step 2: Place the first term in the top left box. 5x2 10 Step 3: Place the third term in the top middle box. Example 4-1a

Step 5: Place the middle term outside the box at the right. Factor In a factor box, the left box times the middle box will equal the right box. Step 4: 5x2 x 10 = 50x2 Step 5: Place the middle term outside the box at the right. Step 6: Find the factors of 50x2 that add to 27x. 5x2 10 50x2 27x Example 4-1a

Factor Step 7: Place the 2 terms that multiply to be 50x2 and add to be 27x in the far right boxes. 5x2 10 50x2 27x 2x 25x Example 4-1a

What can I multiply to get 2x? Factor Before looking at the 4 remaining boxes, remember the rules we have learned thus far: 1.) you multiply from left to right and the product is in the far right box 2.) you multiply from bottom to top and the result is in the top box. Ask yourself: What can I multiply to get 2x? What can I multiply to get 25x? As you are figuring that out, you also have to figure out how to multiply those factors going up to obtain 5x2 and 10. Example 4-1a

Factor Step 8: Place the 2 terms that multiply to be 2x in the top 2 remaining boxes. Step 9: Place the 2 terms that multiply to be 25x in the bottom 2 remaining boxes. 5x2 10 50x2 27x 2x 25x x 2 5x 5 Example 4-1a

Vertical Check: Does x times 5x equal 5x2? Factor Vertical Check: Does x times 5x equal 5x2? Does 5 x 2 = 10? 5x2 27x x 5x 10 2 5 Example 4-1a

Factor Horizontal Check: Does x times 2 equal 2x? Does 5x times 5 = 25x? 27x x 2 2x 5x 5 25x Once everything checks out, you are ready to identify your final answer. Example 4-1a

Factor Your factors are in blue. Circle them diagonally. If there is no sign it is understood to be addition. 27x x 2 2x 5x 5 25x 5x2 10 50x2 Answer: (x + 5)(5x + 2) Example 4-1a

Factor 26x 3x2 35 105x2 x 5 3x 7 5x 21x Answer: Example 4-1a

Factor Look for the factors of 72x2 (24x2 times 3) that add to be -22x. –73 –38 –27 –22 –1, –72 –2, –36 –4, –24 –4, –18 Sum of Factors Factors of 72 The correct factors are –4, –18. Example 4-1b

Factor 24x2 - 22x + 3 Now place the 2 terms that multiply to be 72x2 and add to be -22x in the far right boxes. 24x2 3 72x2 -22x -4x -18x Example 4-1b

Factor 24x2 - 22x + 3 Next: Place the 2 terms that multiply to be -4x in the top 2 remaining boxes. Then: Place the 2 terms that multiply to be -18x in the bottom 2 remaining boxes. 24x2 3 72x2 -22x -4x -18x 4x -1 6x -3 Example 4-1b

Factor 24x2 - 22x + 3 Your factors are in blue. Circle them diagonally. If there is no sign it is understood to be addition. -22x 4x -1 -4x 6x -3 -18x 24x2 3 72x2 Answer: (4x - 3)(6x - 1) Example 4-1b

a. Factor -17x 3x2 10 30x2 x -2 3x -5 -2x -15x Answer: Example 4-1b

b. Factor -23x 10x2 12 120x2 2x -4 5x -3 -8x -15x Answer: Example 4-1b

Example 2 Factor When a, b, and c Have a Common Factor

First factor out the common term. 4(x2 + 6x + 8) Now, factor the trinomial x2 + 6x + 8 Answer: Example 4-2a

First factor out the common term. 2(x2 + 7x + 10) Now, factor the trinomial x2 + 7x + 10 Answer: Example 4-2b

Determine Whether a Polynomial Is Prime Example 3 Determine Whether a Polynomial Is Prime

Factor Look for the factors of -15x2 (3x2 times -5) that add to be 7x. 14 –14 2 –2 –1, 15 1, –15 –3, 5 3, –5 Sum of Factors Factors of –15 Example 4-3a

Answer: is a prime polynomial. There are no factors whose sum is 7. Therefore, cannot be factored using integers. Answer: is a prime polynomial. Example 4-3a

Factor First make a table of the terms that multiply to be 9 but add to be -5. -10 10 -6 6 –1, -9 1, 9 –3, -3 3, 3 Sum of Factors Factors of 9 There are NO such factors so it canNOT be factored. Answer: prime Example 4-3b

Solve Equations by Factoring Example 4 Solve Equations by Factoring

Hint: Key word SOLVE (set equation = 0) Original equation Hint: Key word SOLVE (set equation = 0) Rewrite so one side equals 0. Hint: Use the factor box to factor! -14b 15b2 -8 -120b2 3b 2 5b -4 6b -20b Example 4-4a

Answer: The solution set is Factor the left side. or Zero Product Property Solve each equation. Answer: The solution set is Example 4-4a

Hint: Use the factor box to factor! Solve: 12x2 - 19x + 5 = 0 Original equation 12x2 - 19x + 5 = 0 Hint: Use the factor box to factor! -19x 12x2 5 60x2 3x -5 4x -1 -15x -4x Example 4-4b

Factor the left side. (3x - 1)(4x - 5) = 0 or Zero Product Property Solve each equation. 3x = 1 4x = 5 x = 1/3 x = 5/4 Answer: Example 4-4b

Homework: Page 499 14 - 44 even Use the factor box to assist you with factoring. Example 4-4b