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Algebra I Name Module 3: Quadratic Functions Lessons 4-5 Period Date
Lesson #4: Factoring by Grouping
Day
Today we are going to learn about how to factor by grouping. This will require you to use GCFs twice in the same problem. Sound crazy? It really isnt…
When you see an expression that has FOUR terms, you IMMEDIATELY want to think about factoring by grouping.
Example #1: Factor 5x^3 + 25x^2 + 2x + 10 STEPS
- Check for a GCF
- Split the expression into two groups
- Factor out the GCF from the first group
- Factor out the GCF from the second group
- Do the ‘left overs’ look the same? Because they should!
- Write down the binomial they have in common in one set of parentheses Example #2: Factor x^3 + 2x^2 + 3x + 6 7. Write down the ‘left overs’ as another binomial in a second set of parentheses
- Check your answer by multiplying the two binomials
Worktime: Factor the following expressions by grouping
Lesson #5: Factoring basic trinomials
Now that wasn’t so bad, was it? Good news…we’re going to take a break from factoring by grouping and review some other types of factoring you might find easier. How do we factor basic trinomials? The easiest types of trinomials to factor are ones where the leading coefficient is 1.
Huh?
Let’s review.
A trinomial is a polynomial expression with terms.
A leading coefficient is the that comes first when a polynomial is written in standard form.
Standard form is how you should ALWAYS be writing your polynomial expressions. Standard form is when you write the terms of your expression with the exponents in decreasing order; in other words, from the to the
Try this! Find the product of (x + 7)(x + 3) and write your answer in standard form.
Factoring reverses that process and finds what you can multiply together to get an expression. How could you factor x^2 + 10x + 21?
#1 Factor x^2 + 11x + 24 STEPS
- Write down all the pairs of numbers that multiply to the last #
- Find the pair of #s that add or subtract to give you the middle #
- Draw two sets of parentheses and fill in the #s
- Multiply the binomials to check your answer
x^2 + 11x + 24 is called a expression. That means that the highest power of the variable is 2.
Worktime: Factor the following expressions #2 x^2 + 9x + 14 #3 x^2 + 10x + 16 #4 x^2 + 21x + 20
#5 x^2 + 5x + 6 #6 x^2 + 7x + 6 #7 x^2 + 11x + 30
It is crucial that you are watching the signs when you factor trinomials. Checking your answer is quite easy. Simply multiply the binomials together and see if it matches. You can even check in your calculator if you really want to.
Steps
#8 Factor x^2 – 2x – 24 1.) Write down all the pairs of numbers that multiply to 2.) Determine which pair of numbers can add/subtract to but multiply to 3.) Write out your 2 binomials with the pair of numbers you found 4.) Multiply the two binomials to check your answer WATCH YOUR SIGNS!
#9. Factor c^2 + 2c – 24 #10 Factor x^2 + 15x + 50
#11 Factor b^2 – 10b +24 #12 Factor x^2 – 10x – 24
Day
Today, we are going to continue to look at factoring your basic trinomials. We’re going to look at some tips that might help you factor if you ever get stuck.
Look at the LAST number. If it is negative , the signs are. one ____ and one ____. If it is positive , the signs are the.
If the signs are the SAME
Look at the middle term. BOTH signs will be this sign. EXAMPLES 1.) x^2 + 8x + 12 2.) x^2 + 13x + 42
3.) x^2 – 11x + 30 4.) x^2 – 17x + 70
5.) x^2 + 5x + 4 6.) x^2 – 15x + 50
7.) x^2 – 9x + 18 8.) x^2 – 10x + 9
If the signs are DIFFERENT
Once you figure out what numbers you need, the BIGGER number gets the sign of whatever is on the middle term.
REMINDER! You can ALWAYS check your answer by multiplying the binomials by using distributive property or box method.
EXAMPLES
1.) x^2 – 3x – 18 2.) x^2 – 8x – 20
3.) x^2 + 4x – 12 4.) x^2 + 7x + 12
5.) x^2 – 3x – 40 6.) x^2 – 5x – 14
7.) x^2 – 9x – 10 8.) x^2 – 14x + 40
9.) x^2 + 2x – 24 10.) x^2 – 2x + 1
