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Factoring trinomials are no different than factorization of polynomials except for one additional step.
When factoring trinomials we're looking for the standard form of ax² + bx + c, which will later be used in determining a, b, and c for use in the quadratic formula.
Let us take a look at an example trinomial: x2+16x+32
You can see that this is a trinomial because it consists of three (tri) terms (nominal's) which are x2, 16x, and 32.
Generally, grouping requires us to have 4 terms that either have a common factor for all four terms, or a common factor that is shared between two sets of two terms. i.e. (x² and 16x both have a common factor of x, and y³, 16y², and xy² both have a common factor of y²)
Solving by trinomials by grouping requires a little more mathematical process in order to achieve our factored number. We need, for this problem, a simple mathematical equation:
ac = a+c = b
Meaning that when ever we multiply the number in the a spot and the c spot, the numbers we result in must mathematically add (or subtract) to make b.
In our earlier equation: x2+16x+32 we look at the formula ax² + bx + c.
- a = x (or 1)
- b = + 16
- c = + 32
Always pay attention to the operation signs. When dealing with trinomials there is a huge difference between x2+16x+32 and x2-16x-32.
Looking at our earlier equation: ac = a+c = b, we're saying that 1 • 32 must have some factors that when added equal too 16.
Let's take a look at 32.
- 32 • 1 = 32 but does 31 + 1 = 16? No.
- 16 • 2 = 32 but does 16 + 2 = 16? No.
- 8 • 4 = 32 but does 8 + 4 = 16? No.
Since we've tested this far and have found no numbers that can result in 16, we'd conclude that this problem is Prime. Meaning it is not able to be factored.
Now we'll take a look at a problem that can be factored.
x² - 10x + 21
Again, we need to figure out two numbers that when a • c are multiplied their addition equals to -10. Remember to keep an eye on your signs.
In our case a (1) • c (+21) = 21. Remember when I said we need to pay attention to the signs? This is very important to this problem because a • c needs to equal to -10.
21
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-1 • -21 = -21 but does -1 + -21 = -10? No.
-2, does not go into 21 evenly so we skip this number.
-3 • - 7 = 21 but does -3 + (-7) = -10? Yes.
Now that we've found, what my professor called, "the magic numbers" we need to plug them into our equation, paying attention to the signs we've used.
Our original problem was:
x² - 10x + 21
Now we can replace our "b" with -3 and -7, keeping the variables.
x² - 3x - 7x + 21
Notice anything about our problem now? Now we have the polynomial we need to be able to group and if you remembered so far that a • c = a + c = b, If we take -3x + (-7x), what do we get? -10x, our "b" number.
So we can begin factoring our problem out like any polynomial. We begin by checking to see if there is a GCF (greatest common factor) to our problem. Three of the terms have x's but the fourth does not, two of the terms have 3's as a common factor but the other two do not. So we'd say that the GCF for this problem is 1.
Break the terms down into their smallest components:
- x² = x • x
- -3x = -3 • x
- -7x = -7 • x
- +21 = 3 • 7
Looking at the first two terms x² and -3x, what terms are common to this problem?
- x² = x • x
- -3x = -3 • x
Each term has one "x" variable, this is our "outer number". x(....). We're left with, x and - 3, which is our "inside numbers" x(x - 3)
Now we look at the second two terms, -7x and +21
- -7x = -7 • x
- +21 = 3 • 7
What is common for these two? The 7. So our "outer number" is -7(....). What are we left with? 3 and x. Now, remember we want our "inner numbers" to have the same sign, which if you look at the original broken down form of:
- x • x - 3 • x - 7 • x + 3 • 7
The -7, is going to make 21 a negative when we pull it out. So we simply flip the sign inside the second parentheses from positive to negative.
Our result? 7(x - 3) - x(x - 3).
Look at these problems now as (x+a)(x+b) where (x+a) is our "inner numbers" and (x+b) is our "outer numbers". So inner numbers would be (x - 3) and our outer numbers would be (7 - x), but for mathematical appearance we're flip the outer numbers to look like our inner numbers (x - 7).
So now we're left with our factored trinomial (x - 3)(x - 7).
"But how do you know your answer is correct?" This is a very excellent question.
F.O.I.L, or First, Outer, Inner, Last terms is what we use on factors to check that our solution is correct.
First: (x - 3)(x - 7) = x • x = x²
Outer: (x - 3)(x - 7) = x • -7 = -7x
Inner: (x - 3)(x - 7) = -3 • x = -3x
Last: (x - 3)(x - 7) = -3 • -7 = +21
Put our problem back together: x² - 7x - 3x + 21, combine like terms is x² - 10x + 21.
Congratulations, you've learned how to factor trinomials and check to see if your solution is correct.
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How do you Factor the trinomials of 12x2-32x-12?
12x ^2 -32x-12
When should you use factor by grouping as opposed to regular factoring?
If there are 4 or more terms in a problem, and none are like terms.
What are the rules of factorization?
Do in this order. 1. All, find the gfc 2. Binomial, factor as difference of squares, sum of cubes, difference of cubes. 3. Trinomial, factor as a quadratic. 4. 4 or more terms, factor by grouping.
The grouping method of factoring can still be used when only some of the terms share a common factor A True B False?
True Observe the following: 5x + 5y + 3z = 5(x + y) + 3z The first two terms could be factored because they shared a common factor of 5, but the third term did not -- not all terms need to share a common factor to use the grouping method.
Can you factor the gcf after factoring out the trinomials?
If you've factored out the trinomials and want to find the greatest common factor (GCF) of the remaining terms, you can look for common factors among the coefficients and variables in each term. Let's say you have factored the trinomial � � 2 � � � ax 2 +bx+c into the form � ( � − � ) ( � − � ) a(x−r)(x−s), where � r and � s are the roots or solutions of the trinomial. Now, let's consider the factored form of the trinomial along with any additional terms you have: � ( � − � ) ( � − � ) additional terms a(x−r)(x−s)+additional terms To find the GCF, you'll look for common factors in the coefficients and variables. The GCF will be the product of the common factors. For example, if the remaining terms are 2 � − 4 2x−4, you can factor a 2 from both terms: 2 ( � − � ) ( � − � ) 2 ( � − 2 ) 2(x−r)(x−s)+2(x−2) Now, the GCF is 2 2 because it is the common factor in both terms. If you have specific trinomials or terms you'd like help factoring, feel free to provide them, and I can guide you through the process
What are the kinds of factor?
factoring whole numbers,factoring out the greatest common factor,factoring trinomials,factoring the difference of two squares,factoring the sum or difference of two cubes,factoring by grouping.
How do you do factor by grouping?
Completely factor the expression by grouping 50cp-4wz+5pw-40cz
How do you Factor the trinomials of 12x2-32x-12?
12x ^2 -32x-12
How To Factor Trinomials?
Find one factor by substituting in values, then use long division. You can then apply the quadratic formula to the result - or factorise it by sight, of course
What is the quadratic trinomials factor of 12xsquared-20x -8?
12x2-20x-8 = (12x+4)(x-2)
What is a common factor of the trinomials 2x2- x- 28 and 2x2 - 15x 28?
If that's +28, the answer is (x - 4)
How to factor by grouping?
Common factors are always odd numbers
What are trinomials in algebra?
Trinomials are polynomials with three terms. ie. x2+2x+1
How do you factor perfect square trinomials?
A perfect trinomial must be of the form a2x2 ± 2abxy + b2y2 and this factorises to (ax ± by)2.
Factor by grouping 15v2-19v plus 6?
(5v - 3)(3v - 2)
How do you factor by grouping x3y 2x2y xy3?
xy(x^2 + 2x + y^2)
Factor by grouping 6s'2 plus 29s-42?
(s + 6)(6s - 7)
