In this Grade 9 Saxon Algebra 1 lesson, students learn to factor trinomials of the form ax² + bx + c where the leading coefficient is not equal to 1, using a trial-and-check method to identify the correct binomial factor pairs. The lesson covers cases where b and c are positive, where b is negative, where c is negative, and extends the technique to trinomials with two variables and expressions that require rearranging into descending order before factoring. Students verify their answers using FOIL multiplication as a check.
Section 1
📘 Factoring Trinomials
When a trinomial takes the form , the pattern [for factoring ] no longer works. A new pattern emerges.
Next, you’ll learn the systematic process of testing factor pairs to break down these more complex trinomials into simpler binomials.
Section 2
Factoring With Positive Terms
To factor when and are positive, find pairs of factors for and . All terms in the resulting binomials will be positive.
Think of it as a puzzle! You need factors of 'a' and 'c' that are all positive. Your mission is to arrange them so the 'Outer' and 'Inner' products of FOIL add up to your middle term 'b'.
because the middle term is .
because the middle term is .
Section 3
Example Card: Factoring when b and c are Positive
When all terms are positive, finding the factors is a straightforward treasure hunt. This first key idea, factoring a trinomial where and are positive, simplifies our search.
Example Problem
Factor the expression completely.
Step-by-Step
Section 4
Factoring With a Negative C
To factor when is negative, the last terms in the binomial factors must have opposite signs (one positive, one negative).
When your last term 'c' is negative, it’s a sign showdown! One factor must be positive and one must be negative. You have to test the combinations to see which arrangement gives you the correct middle term 'b'.
since the middle term is .
since the middle term is .
Section 5
Example Card: Factoring when c is Negative
Let's tackle a trinomial where the last term is negative, which introduces a new twist. This second key idea, factoring when is negative, requires checking both positive and negative factor pairs.
Example Problem
Factor the expression completely.
Step-by-Step
Section 6
Factoring With Two Variables
When factoring a trinomial with two variables, like , one variable should descend in power while the other ascends.
Don't panic when you see a second variable! Just treat it like a sidekick. Factor the numbers like normal, then attach the second variable to the last term of each binomial. The FOIL method will work just the same.
because the middle term is .
because the middle term is .
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Section 1
📘 Factoring Trinomials
When a trinomial takes the form , the pattern [for factoring ] no longer works. A new pattern emerges.
Next, you’ll learn the systematic process of testing factor pairs to break down these more complex trinomials into simpler binomials.
Section 2
Factoring With Positive Terms
To factor when and are positive, find pairs of factors for and . All terms in the resulting binomials will be positive.
Think of it as a puzzle! You need factors of 'a' and 'c' that are all positive. Your mission is to arrange them so the 'Outer' and 'Inner' products of FOIL add up to your middle term 'b'.
because the middle term is .
because the middle term is .
Section 3
Example Card: Factoring when b and c are Positive
When all terms are positive, finding the factors is a straightforward treasure hunt. This first key idea, factoring a trinomial where and are positive, simplifies our search.
Example Problem
Factor the expression completely.
Step-by-Step
Section 4
Factoring With a Negative C
To factor when is negative, the last terms in the binomial factors must have opposite signs (one positive, one negative).
When your last term 'c' is negative, it’s a sign showdown! One factor must be positive and one must be negative. You have to test the combinations to see which arrangement gives you the correct middle term 'b'.
since the middle term is .
since the middle term is .
Section 5
Example Card: Factoring when c is Negative
Let's tackle a trinomial where the last term is negative, which introduces a new twist. This second key idea, factoring when is negative, requires checking both positive and negative factor pairs.
Example Problem
Factor the expression completely.
Step-by-Step
Section 6
Factoring With Two Variables
When factoring a trinomial with two variables, like , one variable should descend in power while the other ascends.
Don't panic when you see a second variable! Just treat it like a sidekick. Factor the numbers like normal, then attach the second variable to the last term of each binomial. The FOIL method will work just the same.
because the middle term is .
because the middle term is .
Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter