Warm Up Find each product. 1. (x – 2)(2x + 7) 2. (3y + 4)(2y + 9) 3. (3n – 5)(n – 7) Find each trinomial. 4. x2 +4x – 32 5. z2 + 15z + 36 6. h2 – 17h + 72 2x2 + 3x – 14 6y2 + 35y + 36 3n2 – 26n + 35 (x – 4)(x + 8) (z + 3)(z + 12) (h – 8)(h – 9)

Warm Up  

Objective Factor quadratic trinomials of the form ax2 + bx + c.

Find a pair of numbers that multiply to the first number, and add to the second. Multiplies to 24; Adds to 10 7. Multiplies to 36; Adds to 13 6 & 4 9 & 4 2. Multiplies to -8; Adds to -2 8. Multiplies to -19; Adds to -18 -4 & 2 -19 & 1 3. Multiplies to 15; Adds to 8 9. Multiplies to 6; Adds to -5 5 & 3 -3 & -2 4. Multiplies to 21; Adds to -10 10. Multiplies to -14; Adds to -5 -7 & -3 -7 & 2 5. Multiplies to -27; Adds to 6 11. Multiplies to 16; Adds to 10 9 & -3 8 & 2 6. Multiplies to -49; Adds to 0 12. Multiplies to 15; Adds to 16 7 & -7 15 & 1

So, to factor a2 + bx + c, check the factors of a and the factors of c in the binomials. The sum of the products of the outer and inner terms should be b. Product = a Product = c Sum of outer and inner products = b ( X + )( x + ) = ax2 + bx + c

The guess and check method is usually not the most efficient method of factoring a trinomial. Look at the product of (x + 3) and (x + 4). x2 12 (x + 3)(x +4) = x2 + 7x + 12 3x 4x The coefficient of the middle term is the sum of 3 and 4. The third term is the product of 3 and 4.

 

Example 2A: Factoring ax2 + bx + c When c is Positive Factor each trinomial. Check your answer. 2x2 + 17x + 21 a = 2 and c = 21, Outer + Inner = 17.   ( x + )( x + )   Two numbers that multiply to 42; Add to 17: 14 & 3 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: 2x GCF of 2nd group: 3      

When b is negative and c is positive, the factors of c are both negative. Remember!

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 + 6x + 5 a = 1 and c = 5, Outer + Inner = 6.   ( x + )( x + )   Two numbers that multiply to 5; Add to 6: 5 & 1 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: x GCF of 2nd group: 1      

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 2x2 + 9x - 18 a = 2 and c = -18, Outer + Inner = 9.   ( x + )( x + )   Two numbers that multiply to -36; Add to 9: 12 & -3 Break up the b term into the two numbers that you found.   Factor by grouping; GCF of 1st group: 2x GCF of 2nd group: -3        

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 3x2 - 16x + 16 a = 3 and c = 16, Outer + Inner = -16.   ( x + )( x + )   Two numbers that multiply to 48; Add to -16: -12 & -4 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: 3x GCF of 2nd group: -4      

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 4x2 - 15x - 4 a = 4 and c = -4, Outer + Inner = 13.   ( x + )( x + )   Two numbers that multiply to -16; Add to -15: -16 & 1 Break up the b term into the two numbers that you found.   Factor by grouping; GCF of 1st group: 4x GCF of 2nd group: 1        

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 - 8x + 15 a = 1 and c = 15, Outer + Inner = -8.   ( x + )( x + )   Two numbers that multiply to 15; Add to -8: -5 & -3 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: x GCF of 2nd group: -3      

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 6x2 + 17x + 5 a = 6 and c = 5, Outer + Inner = 17.   ( x + )( x + )   Two numbers that multiply to 30; Add to 17: 15 & 2 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: 3x GCF of 2nd group: 1      

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 - 5x + 6 a = 1 and c = 6, Outer + Inner = -5.   ( x + )( x + )   Two numbers that multiply to 6; Add to -5: -2 & -3 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: x GCF of 2nd group: -3      

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 3x2 + 11x - 4 a = 3 and c = -4, Outer + Inner = 13.   ( x + )( x + )   Two numbers that multiply to -12; Add to 11: 12 & -1 Break up the b term into the two numbers that you found.   Factor by grouping; GCF of 1st group: 3x GCF of 2nd group: -1        

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 9x2 - 15x + 4 a = 9 and c = 4, Outer + Inner = -15.   ( x + )( x + )   Two numbers that multiply to 36; Add to -15: -12 & -3 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: 3x GCF of 2nd group: -1      

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. 3x2 + 13x + 12 a = 3 and c = 12, Outer + Inner = 13.   ( x + )( x + )   Two numbers that multiply to 36; Add to 13: 9 & 4 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: 3x GCF of 2nd group: 4      

Example 2A: Factoring ax2 + bx + c Factor x2 + 10x + 21. Show that the original polynomial and the factored form have the same value for n = 0, 1, 2, 3, and 4. x2 + 10x + 21 a = 1 and c = 21, Outer + Inner = 10.   ( x + )( x + )   Two numbers that multiply to 21; Add to 10: 7 & 3 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: x GCF of 2nd group: 3      

Example 4A Continued Evaluate the original polynomial and the factored form for n = 0, 1, 2, 3, and 4. (y + 7)(y + 3) (0 + 7)(0 + 3) = 21 (1 + 7)(1 + 3) = 32 (2 + 7)(2 + 3) = 45 (3 + 7)(3 + 3) = 60 (4 + 7)(4 + 3) = 77 y 1 2 3 4 y2 + 10y + 21 02 + 10(0) + 21 = 21 y 1 2 3 4 12 + 10(1) + 21 = 32 22 + 10(2) + 21 = 45 32 + 10(3) + 21 = 60 42 + 10(4) + 21 = 77 The original polynomial and the factored form have the same value for the given values of n.

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 - 9 a = 1 and c = -9, Outer + Inner = 0.   ( x + )( x + )   Two numbers that multiply to -9; Add to 0: -3 & 3 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: x GCF of 2nd group: 3      

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 - 36 a = 1 and c = -36, Outer + Inner = 0.   ( x + )( x + )   Two numbers that multiply to -36; Add to 0: -6 & 6 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: x GCF of 2nd group: 6      

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 - 81 a = 1 and c = -81, Outer + Inner = 0.   ( x + )( x + )   Two numbers that multiply to -81; Add to 0: -9 & 9 Break up the b term into the two numbers that you found.     Factor by grouping; GCF of 1st group: x GCF of 2nd group: 9      

Example 2A: Factoring ax2 + bx + c Factor each trinomial. Check your answer. x2 + 25     Two numbers that multiply to 25; Add to 0: …..?   THE KEY to this problem is having c be a negative number, not a positive

When the leading coefficient is negative, factor out –1 from each term before using other factoring methods.

When you factor out –1 in an early step, you must carry it through the rest of the steps. Caution

Factor. Check your answer.