Polynomials & Factoring

Polynomials are just expressions with variables raised to exponents, added or subtracted. The most common type we work with in Algebra 1 are binomials (two terms). It is important to know the following vocabulary when discussing polynomials.
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Polynomials refer to any term or combination of terms where the variables have positive, whole number exponents. For more information on polynomials, watch this video:

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These are examples of polynomials:

x², 2x+2, 3x²+5y², -12y3

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These are examples of non-polynomials:

√x , 1x

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The variables must not be in the denominator or in a root.

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Multiplying Binomials:

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When you multiply two binomials, you’ll use the FOIL rule:

• F = First terms
  
• O = Outer terms
  
• I = Inner terms
  
• L = Last terms
  

This ensures you don’t miss anything when you expand.

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Example 1: Multiply (4x – 7)(2x + 3)

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Step 1: Apply FOIL

• First: (4x)(2x) = 8x²
  
• Outer: (4x)(3) = 12x
  
• Inner: (–7)(2x) = –14x
  
• Last: (–7)(3) = –21

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Step 2: Add them together
8x² + 12x – 14x – 21

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Step 3: Combine like terms
12x – 14x = –2x

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So the result is: 8x² – 2x – 21

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Here is another example:

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Example 2: Multiply (5x + 2)(x – 6)

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Step 1: FOIL again

• First: (5x)(x) = 5x²
  
• Outer: (5x)(–6) = –30x
  
• Inner: (2)(x) = +2x
  
• Last: (2)(–6) = –12

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Step 2: Put it all together
5x² – 30x + 2x – 12

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Step 3: Combine like terms
–30x + 2x = –28x

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Final Answer: 5x² – 28x – 12

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Factoring Polynomials

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Factoring is the opposite of multiplying. You are trying to find the common factors between terms. We can factor using different methods: factor out the GCF, factor by grouping, or factor by completing the square.

For more information regarding factoring polynomials, watch this video as a walkthrough:

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Reviewing quadratics can help with this!

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Example 3: Factor 6x² + 11x – 10

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We’ll use the completing the square method (or the AC method).

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Step 1: Identify a, b, c
Here we have 6x² + 11x – 10
So: a = 6, b = 11, c = –10

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Step 2: Multiply a × c
6 × –10 = –60

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Step 3: Find two numbers that multiply to –60 and add to 11
Those are +15 and –4

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Step 4: Split the middle term
6x² + 15x – 4x – 10

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Step 5: Group terms
(6x² + 15x) – (4x + 10)

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Step 6: Factor each group
3x(2x + 5) – 2(2x + 5)

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Step 7: Factor out the common binomial
(3x – 2)(2x + 5)

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Final Answer: (3x – 2)(2x + 5)

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Teacher Tips

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• FOIL is just distributive property with a shortcut name. You’re multiplying each term in the first set of brackets by each term in the second.
  
• Always combine like terms after multiplying. That’s the cleanup step many students skip.
  
• When factoring, if the leading coefficient (a) isn’t 1, use the AC method — it keeps things organized.
  
• Check factoring by multiplying back out. If you don’t get the original, fix your steps.
  
  

Quick Practice Problems

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Try these (answers below):

1. (x + 4)(x + 7)
   
2. (2x – 5)(3x + 1)
   
3. Factor x² – 9x + 20
   
4. Factor 2x² + 7x + 3
   

Answers:

1. x² + 11x + 28
   
2. 6x² – 13x – 5
   
3. (x – 5)(x – 4)
   
4. (2x + 1)(x + 3)
   
   

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