Factoring polynomials: special cases

Factoring polynomials: special cases

 $x^2\ +\ bx\ +\ c$  Factoring Review

• In  the last lesson we looked at factoring by the grouping method.  This method can also be used to factor trinomials. 

• Ex.   $x^2\ -7x\ +10$  

• Set up a table:  

• Set up a Table:   Factors of +10 $\perp$  Addend of -7

• Factors of + 10: 1 x 10, 2 x 5, -1 x -10, -2 x -5

• Now of those listed which will add up to -7.

• Only -2 x -5, because - 2 + -5 = -7

Now put those values into the middle term of the original equation

 $x^2\ -\ 2x\ -5x\ +\ 10$  you now have 4 terms which can be factored by grouping.
 $\left(x^2\ -2x\right)\ +\ \left(-5x\ +\ 10\right)$  
x(x - 2) -5(x - 2)
(x-2)(x-5) is the final answer.

Multiple Choice

Factor
k²+7x+10

• (k+2)(k+5)
• (k-2)(k-5)
• (k+10)(k+4)
• (k+2)(k-5)

Multiple Choice

Factor
n²-13n+40

• (n-5)(n-8)
• (n+5)(n-8)
• (n-5)(n+8)
• (n+6)(n+1)

Multiple Choice

Factor
k² -2k - 24

• (k+4)(k-6)
• (k+4)(k+6)
• (k+6)(k-1)
• (k-4)(k+6)

Multiple Choice

Factor
n² + 16n + 63

• (n-7)(n+4)
• (n+7)(n-9)
• (n-3)(n-10)
• (n+7)(n+9)

Factoring when the lead coefficient is not = to 1

• The same methods can be used to factor a trinomial that the lead coefficient is greater than 1

•  $8b^2\ +\ 18b\ +\ 9$ is the first example

  
  

• Again set up the factors and addends table.

• Factors will now be for 72 because you multiply 8 x 9.  They will have to add to 18.

• Factors of 72.  Since both signs are positive, there is no need to worry about the integers.

• Factors of 72: 1 x 72, 2 x 36, 3 x 24, 4 x 18, 6 x 12, 8 x 9

• From the list of factors, which will add up to a + 18?

• Looks like 6 x 12 because 6 + 12 = 18.

• Now put it into the original equation and split the middle term with those factors.

• 
  
   $8b^2\ +\ 6b\ +\ 12b\ +\ 9$  
  

 $Group\ the\ terms:\ \left(\ 8b^2\ +\ 6b\right)+\left(12b\ +\ 9\right)$  

Factor: 2b( 4b + 3) + 3b(4b + 3)

(4b + 3)(2b + 3b) is the factored answer.

Multiple Choice

Factor:
 2m² + 3m - 9

• 2(m + 3)(m - 3)
• (2m - 3)²
• (m + 3)(2m - 3)
• (2m + 3)(m - 3)

Multiple Choice

Factor:
 4h² - 17h + 4

• (2h - 2)²
• 4(h + 4)(h + 1)
• (2h - 2)(2h + 2)
• (h - 4)(4h - 1)

Multiple Choice

Factor:
 2m² + 3m - 9

• 2(m + 3)(m - 3)
• (2m - 3)²
• (m + 3)(2m - 3)
• (2m + 3)(m - 3)

Multiple Choice

Factor Completely:  6x² + 5x - 6

• 6(x + 1)(x - 6)
• (6x + 6)(x - 1)
• (2x - 3)(3x + 2)
• (3x - 2)(2x + 3)

Multiple Choice

Factor Completely:  3n² - 15n + 18

• (n - 2)(n - 3)
• 3(n + 2)(n + 3)
• (3n - 9)(n - 2)
• 3(n - 3)(n - 2)

Special cases: factoring polynomials

• Factoring the difference of squares:

   $a^2\ -\ 49$  we did something similar in chapter 5 see if you recognize it.

• (a - 7)(a + 7) is the factored answer

• Do you recognize it from (x -y)(x_y) when we multiplied polynomials.

• If the binomial cannot be factored  $x^2\ +\ 16$  then we say it is prime.

Perfect square trinomials

• Ex.

   $16a^2-56a=49$  

• Perfect square of  $16a^2$   is 4a; perfect square of 49= 7 the middle terms sign will be the sign in the answer  $\left(4a\ -7\right)^2$  

• No need to work overtime in factoring know the formula.

• Ex.   $16r^2+\ 24rm\ +\ 9m^2$  

• square root of  $16r^2$   is 4r ;  $9m^2$   is 3m so  $\left(4r\ +\ 3m\right)^2$  is the factored answer.

Multiple Choice

Factor:
x² + 81

• Prime (not factorable)
• ( x - 9 ) ( x + 9 ) 
• ( x + 9 ) ( x + 9 ) 
• ( x - 9 ) ( x - 9 ) 

Multiple Choice

Factor:  $16y^2-24y+9$  

•  $\left(4y+3\right)\left(4y-3\right)$  

•  $\left(8y\ +\ 3\right)^2$  

•  $\left(8y-3\right)^2$  

•  $\left(4y-3\right)^2$  

Multiple Choice

 $16x^2-81$  Factor:

•  $\left(4x-9\right)^2$  

•  $\left(2x-9\right)^2$  

•  $\left(4x\ +\ 9\right)^2$  

•  $\left(4x+9\right)\left(4x-9\right)$  

Multiple Choice

 $a^2-10a+25$  Factor:

•  $\left(a-5\right)\left(a+5\right)$  

•  $\left(a+5\right)^2$  

•  $\left(a-5\right)^2$  

•  $\left(a-10\right)^2$  

Multiple Choice

 $64x^2+112x+49$  Factor:

•  $\left(8x-7\right)\left(8x+7\right)$  

•  $\left(8x-7\right)^2$  

•  $\left(16x+7\right)^2$  

•  $\left(8x+7\right)^2$  

Multiple Choice

Factor:  $x^2-121$  

•  $\left(x-11\right)^2$  

•  $\left(x+11\right)\left(x-11\right)$  

•  $\left(x-60.5\right)\left(x+60.5\right)$  

•  $\left(x+11\right)^2$  

• Conclusion: factoring is not that difficult.

• 1. First before you begin see if there are GCF's that can be factored first.

• Then which pattern are you factoring?

• Is it a special case?

• These should be the questions you ask yourself as you move on to the homework that is due on December 1. Have a great break! Stay Healthy.