Factoring trinnomials with a lead coefficient of 1
Description/Explanation/Highlights
Video Description
This video explains how to factor trinomials when the lead coefficient is one.
Steps and Key Points to Remember
To factor a trinomial with a lead coefficient of 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 \(x^2+5x+6\) there are exactly three terms: \(x^2\), \(5x\), and \(6\).
- In this lesson we will focus on trinomials with a lead coefficient of 1, meaning theat there is no number in front of \(x^2\) implying that it is one.
- Trinomials factor into two binomials (two terms) with two sets of parenthesis. i.e. \(\text{( ) ( )} \)
- 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 original trinomial. What two numbers when multiplied together will give you the last number and in our example, since the sign is positive, when added together will give us the number before x in the middle term? Note: If the sign of the last term had been negative, we would have subtracted to get the middle terms. In our example \(3 \cdot 2 = 6\) and \(3+2=5\)
- 3 is the second term of the first binomial and 2 is the second term of the second binomial although which binomial gets 3 and which gets 2 does not really matter. i.e. \( \text{(x 3)(x 2) or (x 2)(x 3)}\)
- Since the sign of the third term was positive and we added, both signs in the binomial will be the same and the sign of the second term of the trinomial tells us what it will be, in this example use “+” since the number in front of x is +5. In this example, the factors would be (x + 3)(x + 2)
- A good way to double check is to multiply the outside terms together and the inside terms together and then add them to see if it equals the original middle term in the trinomial. In our example: multiple x \(\cdot\) 2 to get 2x and then 3 \(\cdot\) x to get 3x which when added together will equal 5x; the middle term of the original trinomial.
- In the example: \(x^2-9x+20\) we still multiply to get 20 and add to get 9 so the numbers are 4 and 5 but since the sign of the middle term is negative, both signs are negative. The factored version is: (x – 5)(x – 4). Notice how multiplying the outside and inside terms together and adding them will bet -9x this time as it should.
- What if the sign of the third term is negative? Then we subtract the numbers to find the middle number. For example: \(x^2+x-20\) has a “-” sign in front of the last term 20. This time we look for two numbers that multiply together to equal twenty but subtract to equal one (the number implied to be in front of x). 5 x 4 = 20 and 5 – 4 = 1 so 5 and 4 are the second terms of the binomial factors but since we subtracted this time, the signs must be opposite. The sign of the middle term of the trinomial tells us the sign of the big number in the binomial. Since the sign of the middle term in our example is positive, 5 get a “+” sign and 4 gets a “-” sign. The final factored result is (x + 5)(x – 4). Notice when you multiply x by -4 you get -4x and 5 by x get +5x. Added together, the result is x, the value of the middle term.
- If the sign of the middle term had also been negative, the big number would have been negative. In the example: \(x^2 -4x-12\), 4x has a “-” sign and there is also a “-” in front of the last term, 12. This time we look for two numbers that multiply together to equal 12 but subtract to equal 4. 6 x 2 = 12 and 6 – 2 = 4 so 6 and 2 are the second terms of the binomial factors but since we subtracted, the signs must be opposite. The sign of the middle term of the trinomial tells us the sign of the big number in the binomials. The sign of the middle term in this example is negative, so since 6 is greater than 2, 6 gets a “-” sign and 2 gets a “+” sign. The final factored result is (x – 6)(x + 2).
Here are some key points to keep in mind when factoring a trinomial with a lead coefficient of 1:
- Not all trinomials factor.
- Trinomials that can be factored will always factor into two binomials.
- If the lead coefficient is 1, then each binomial factor will have x as the first term.
- The sign of the third term in the trinomial determines whether its numberic factors will be added (if “+”) or subtracted (if “-“) to get the middle term’s coefficient.
- If added, the sign of the middle term determines what both signs will be (always the same). Both signs will be the same as the sign of the middle term in the original trinomial.
- If subtracted, the middle sign will determine the sign of the biggest numeric factor in the binomials and the two signs will always be opposite.
- A good way to double check work is to multiply the outside terms together and the inside terms together then combine the two to be sure it equals the original middle term.
Video Highlights
- 00:00 \(x^2+5x+6\) factoring example with signs the same (positive)
- 02:20 \(x^2-9x+20\) factoring example with first sign negative and second sign positive
- 03:30 \(x^2+x-20\) factoring example with first sign positive and second sign negative
- 05:25 \(x^2-4x-12\) factoring example with both signs negative
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