Factoring trinomials recap

Factoring trinomials recap

An optional tutoring lesson on factoring that should help fill in the gaps.

Let's start with a review of Multiplication

• If you have two binomials (x + a) and (x + b) ...

• When you multiply these together the result is 4 terms: (2 terms)(2 terms) = 4 terms

• Use the Area Model for this multiplication.

1st Step

• Draw a 2 x 2 table

• Place one factor on the top of the table

• Place the other factor on the left side of the table

2nd Step

• Find the "area" of each cell by multiplying the edges.

• Top left is x · x, this equals x²

• Top right is a · x, this equals ax

• Bottom left is b · x, this equals bx

• Bottom right is a · b, this equals ab

3rd Step

• Add all the 'areas'

• Then you would combine like terms (ax + bx) = (a + b)x

• Rewrite the trinomial

•  $x^2\ +\ \left(a+b\right)x\ +\ ab$  

• Since a & b are numbers, we can see the middle terms is the SUM of the consants and the last term is the PRODUCT of the same numbers

Example:

• Let a = 2 and b = -3

• Then we multiply (x + 2)(x - 3)

• Set up Area Model

Example:

• Find the area of each cell

• Write out the SUM of the cells

• Combine like terms

• Final result is a trinomial

Multiple Choice

Multiply (x + 1)(x + 4)

•  $x^2-5x-4$  

•  $x^2+5x+4$  

•  $x^2-4x-5$  

•  $x^2+4x+5$  

• no correct answer

Multiple Choice

Multiply (x - 1)(x - 4)

•  $x^2-5x+4$  

•  $x^2+5x-4$  

•  $x^2-4x+5$  

•  $x^2-4x+5$  

• no correct answer

Multiple Choice

Multiply (x - 1)(x + 4)

•  $x^2-5x-4$  

•  $x^2+5x+4$  

•  $x^2-3x-4$  

•  $x^2+3x-4$  

• no correct answer

Multiple Choice

Multiply (x + 1)(x - 4)

•  $x^2-5x-4$  

•  $x^2+5x+4$  

•  $x^2+3x-4$  

•  $x^2-3x+4$  

• no correct answer

NOW, Let's turn it around with factoring...

• Factoring is the opposite operation from multiplication

• Factoring divides the polynomial into a product 2 or more prime factors.

• For example, if we multiply (x + 1)(x + 2), we get

   $x^2+3x+2$  

• So factoring does the opposite. We start with 

   $x^2+3x+2\ \rightarrow\left(x+1\right)\left(x+2\right)$  

• REMEMBER that the standard for is  $ax^2+bx+c$  where a, b, & c are REAL numbers.

Factoring using the AREA Model

• Remember, when we multiply, we place the 2 binomials outside on the top and left.

• In factoring, we start with the first term inside top left.

• Then place third term inside bottom right.

• Let's factor x^2 + 3x + 2

2nd step - SUM & PRODUCT

• What 2 number SUM to 3 and their PRODUCT is 2?

•  $2\times1=2$  

•  $2+1=3$  

• Place these values inside the two remaining cells. (Order doesn't matter.)

3rd Step (1st part):

• Factor GCF out of top row

•  $x^2+2x\rightarrow x\left(x+2\right)$  

• Write X + 2 on the top of the model.

3rd Step (2nd part):

• Factor GCF out of left column

•  $x^2+x\rightarrow x\left(x+1\right)$  

• Write X + 1 on the left side of the model.

• SO, our factors for this trinomial are (x + 2)(x + 1 ) OR (x + 1)(x + 2)

Multiple Choice

Factor

 $x^2+6x+8$  

•  $\left(x+5\right)\left(x+1\right)$  

•  $\left(x+2\right)\left(x+3\right)$  

•  $\left(x+2\right)\left(x+6\right)$  

•  $\left(x+4\right)\left(x+2\right)$  

Multiple Choice

Factor

 $x^2-6x+8$  

•  $\left(x-8\right)\left(x-1\right)$  

•  $\left(x-2\right)\left(x-4\right)$  

•  $\left(x+2\right)\left(x-4\right)$  

•  $\left(x+4\right)\left(x-2\right)$  

Multiple Choice

Factor

 $x^2-2x-8$  

•  $\left(x-8\right)\left(x+1\right)$  

•  $\left(x-2\right)\left(x-4\right)$  

•  $\left(x+2\right)\left(x-4\right)$  

•  $\left(x+4\right)\left(x-2\right)$  

Multiple Choice

Factor

 $x^2+2x-8$  

•  $\left(x-8\right)\left(x+1\right)$  

•  $\left(x-2\right)\left(x-4\right)$  

•  $\left(x+2\right)\left(x-4\right)$  

•  $\left(x+4\right)\left(x-2\right)$  

NEXT

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