Factoring trinnomials with a lead coefficient other than 1
Description/Explanation/Highlights
Video Description
This video explains how to factor trinomials when the lead coefficient is not one.
Steps and Key Points to Remember
To factor a trinomial with a lead coefficient other than 1, follow these steps:
- Make sure the expression you are factoring is a trinomial by counting the terms separated by addition or subtraction signs. Trinomials must have exactly three. In the example: \(12x^2-x-6\) there are exactly three terms: \(12x^2\), \(-x\), and \(-6\)
- In this lesson we will focus on trinomials with a lead coefficient greater than 1, meaning that there is a number in front of \(x^2\)
- We will focus on one method used to factor when there is a lead coefficient. There are several methods to accomplish this but this one seems to lead to the fewest errors in my experience. The name often given to this technique is “bottoms up.”
- trinomials factor into two binomials (two terms) with two sets of parenthesis. i.e. \(\text{( )( )}\)
- Begin by multiplying the lead coefficient by the last term in the trinomial and making the lead coefficient one (effectively dropping the lead coefficient). In the example above, multiply 12 by -6 resulting in a new trinomial of \(x^2-x-72\).
- Now factor this new trinomial in the same way you would any trinomial with a lead coefficient of 1.
- Find the square root of \(x^2\) which will always be x.
- x is the first term of both binomials. i.e. \(\text{(x )(x )}\)
- Look at the middle and last terms along with their signs in the new trinomial. What two numbers when multiplied together will give you the last number and in our example, since the sign is negative, subtract to give us the number before x in the middle term? In our example: \(9\cdot8=72\) and \(9-8=1\)
- 9 is the second term of the first binomial and 8 is the second term of the second binomial although which binomial gets 9 and which gets 8 does not really matter. i.e. \(\text{(x 9)(x 8)}\) or \(\text{(x 8)(x 9)}\)
- Since the sign of the third term was negative and we subtracted, the signs in the binomials will be opposite and the sign of the second term of the trinomial tells us the sign of the big number. In this example, it is also negative so the big number (9) gets the negative sign. In this example, the factors would be: \((x-9)(x+8)\)
- If the original problem had been a lead coefficient of 1, we would be finished but don’t forget that we multiplied the lead coefficient by the last term so we must now “reverse” that process.
- Divide the second term in each factor by the original coefficient.
- In the example it would be: \((x-\frac{9}{12})(x+\frac{8}{12})\)
- REDUCE the fractions if they will reduce. THIS IS VERY IMPORTANT in getting the correct answer!
- The new factors in our example will now look like this: \((x-\frac{3}{4})(x+\frac{2}{3})\)
- Now, here is where the name “bottoms up” comes from. If there is still a number in the denominator of either of the two fractions, move that denominator from the bottom up to the front of x. In our exammple there is a 4 in one binomial and a 3 in the other. Move them in front of x in their respective binomals. These are the final factors. \((4x – 3)(3x + 2)\)
- If reducing the fraction causes the denominator to become one and in effect go away, there is no number to move up and that factor is complete.
Here are some key points to keep in mind when factoring a trinomial with a lead coefficient other than 1:
- Not all trinomials factor even after multiplying the lead coefficient by the last term.
- Any new trinomial that can be factored will always factor into two binomials.
- If the lead coefficient is greater than 1, then multiply it by the third term and drop the lead coefficient.
- Factor the new trinomial like any trinomial with a lead coefficient of one (see related video).
- After factoring divide the second term in both binomial factors by the original lead coefficient.
- REDUCE the fractions if they can be reduced! You must do this to get the correct factors!
- Move any remaining denominators to the front of x in each factor. This will be the factored version.
- If reducing a fraction does not leave a fraction, that factor is complete.
Video Highlights
- 00:00 \(12x^2-x-6\) factoring example with lead coefficient of 12 and both signs negative
03:00 \(2x^2-3x-20\) factoring example with lead coefficient of 2 and both signs negative
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