TEN POINTS for best answer! Trinomials?

Question

Please show work, I'd love to learn something, not just get the right answer.

24k² + k - 3

25w² - 5w - 6

m² - 18m - 40

Answer 1

(3k-1)(8k+3)

(5w-3)(5w+2)

(m+2)(m-20)

To solve these, follow these steps examples will be for the first trinomial):

1. Determine the possible factors for the coefficient of the first term e.g. 24 - 1*24, 2*12, 3*8, 6*4.

2. Determine the possible factors for the third term e.g. 3 has only 1*3.

3. Arrange these in such a way that it will result in the coefficient of second term remembering to account for the sign of the third term. This takes some trial and error, but eventually you will develop a sense for it.

Answer 2

Hi,

I'll show all of the work on the first one, and you can use it as an example for the others.

I've always liked to look for factoring possibilities first. The possible combinations for the first coefficient are:

(2, 12)
(3, 8)
(4,6)

And for the second are:
(1,3)

Since we know that we need to have (a + b) (c + d) so that a*d + b*c = 1 in this case (so that we can get the one k for the center term in the original expression). I think we should try (3,8) and (1,3) for our combination because then (when we adjust the signs) we'll get 3*3 + (-8)*1 = 1 or 3*3 + 8*(-1) Which is what we want.

So:
24k^2 + k - 3 = (3k 1) (8k 3)

now we just need the signs to make it make sense. Since the third term in the original expression needs to be negative, we'll need opposite signs. Also, since the second term in the original expression is postive, the 3 needs to be positive so that we'll have 9k-8k instead of the other way around.

so

24k^2 + k - 3 = (3k - 1) (8k + 3)

I always multiply it out to check my work, and it works. (Unless I'm dumb today, so you'd better check it too)

Assuming that you'd like to find the roots of this expression instead of just a factorization (because really, why do it if you're not going to solve for the answer).

The roots are
(3k - 1) (8k + 3) = 0

so either
3k-1 = 0
3k = 1
k =1/3

or
8k +3 = 0
8k = -3
k = -3/8

QED

Hope that helps,
Matt

Answer 3

1) 24k² + k - 3

- (do the cross multiply thing by putting the factors of the squared part on the left and the -3)
8k + 3 = 9k (cross multiplied 3k by 3 and 8k by -1)
3k -1 = -8k
- when you subtract 9k and 8k (since 8k is negative but if not, you should add. it depends on the operations) you will get k which is the middle value of a trinomial (which is k in this equation) .this is how you test if you factored correctly. try other factors of the numbers if you did not arrive at the correct value since this is a trial and error thing.
- combine the values that you have created and there is your answer
(8k + 3) (3k - 1)

now answer the succeeding numbers in the same manner

** you can do this mentally. this is just another way to solve

Answer 4

I am not providing you any answers but I'll give some guidance how to factor these that you noted. Basically, we are reversing the FOIL method to get our factored form. We are looking for two binomials that when you multiply them you get the given trinomial.
Step 1: Set up a product of two ( ) where each will hold two terms. It will look like this: ( )( ). Step 2: Find the factors that go in the first positions. To get the x squared (which is the F in FOIL), we would have to have an x in the first positions in each ( ). So it would look like this: (x )(x ).
Step 3: Find the factors that go in the last positions.The factors that would go in the last position would have to be two expressions such that their product equals c (the constant) and at the same time their sum equals b (number in front of x term).As you are finding these factors, you have to consider the sign of the expressions:If c is positive, your factors are going to both have the same sign depending on b’s sign.
If c is negative, your factors are going to have opposite signs depending on b’s sign.Factor the trinomial . Note that this trinomial does not have a GCF. So we go right into factoring the trinomial of the form .Step 1: Set up a product of two ( ) where each will hold two terms. It will look like this: ( )( ) Step 2: Find the factors that go in the first positions.
Since we have y squared as our first term, we will need the following: (y )(y )Step 3: Find the factors that go in the last positions. We need two numbers whose product is 6 and sum is -5. That would have to be -2 and -3.Putting that into our factors we get: *-2 and -3 are two numbers whose prod. is 6 and sum is -5. Note that if we would multiply this out, we would get the original trinomial. For more information visit www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut28_facttri.htm

Answer 5

I would use the quadratic on this one

x = 1 +-Square root of 1^2 - 4(24)(-3)/2(24)
x = 1 +-Square root of 1+288/48
x = 1 +-Square root of 289/48
x = 1 +-17/48
x = 48/48 + 17/48 and/or x = 48/48 - 17/48
x = 65/48 and/or x = 31/48

Answer 6

24k^2 +k -3
(8k + 3)(3k - 1)

25w^2 -5w - 6
(5w - 3)(5w + 2)

m^2 -18m -40
(m - 20)(m + 2)

Answer 7

24k² + k - 3
24k² +9 k-8k - 3
24k² +9 k-8k - 3
3k(8k+3)-1(8k+3)
(8k+3)(3k-1)

25w² - 5w - 6
25w² -15w+10w - 6
5w(5w -3)+2(5w -3)
(5w -3)(5w +2)

m² - 18m - 40
m² - 20m+2m - 40
m(m - 20)+2(m - 20)
(m - 20)(m + 2)