Polynomials & Factoring: Terms, Binomials & FOIL

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):

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

Answers:

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

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