Questions About How To Factor Trinomials

Mark asks…

how do you factor trinomials quickly and in your head?

I'm in an upper level math class and am fed up with having to factor on paper every time. In class ive noticed some students can factor trinomials even with leading coefficients of two in a second or two. Is there a secret?

admin answers:

Factor the first coefficient.
Factor the last coefficient.

Then find the product that results in the middle coefficient.

It's especially helpful if one of the numbers is prime and has only one possible way of factoring (technically 2 if you count negatives). Also, if you get the right number, but the opposite sign, just flip them around.

It's just practice and experience.

Example:
6x² + 11x - 10

Choices for the first coefficient are: 1 x 6 or 2 x 3
Choices for the last coefficient are: 1 x -10 or 2 x -5 (or reversed).

You are going to have different signs at the end, so you need something bigger than 11 which you will subtract from to get 11.

Basically it is trial and error, but in your head.
1 x 6, 1 x -10 <--- no good
1 x 6, 2 x -5 <-- no good
2 x 3, 1 x -10 <-- no good
2 x 3, 2 x -5 <-- no good
2 x 3, 5 x -2 <-- GOOD

(2x + 5)(3x - 2)

Joseph asks…

How do I factor trinomials on my calculator?

I have a TI-84 Plus Silver edition and need to factor problems such as:

x^2 - 14xy + 24y^2

Is there a program i can download or a specific way to do this type of factoring

admin answers:

X² - 14xy + 24y²
= x² - 14xy + 49y² - 25y²
= (x)² - 7.2.x.y + (7y)² - (5y)²
= (x - 7y)² - (5y)²
= (x - 7y - 5y)(x - 7y + 5y)
= (x - 12y)(x - 2y)

Carol asks…

How do you factor trinomials with lead coefficients other than one?

I have factoring homework for my Algebra 2 class. I remembered everything but this section, so I need help. I know that the steps you have to start with is a*c= whatever and b = whatever then the two numbers that multiple to equal the ac and add for the b, but I'm lost after that. The ones I can't get are 9a^2-6a+1, 6x^2-7x+2, and 2x^2+13x-7... Any help is greatly appreciated.

admin answers:

9a² - 6a + 1 = (3a - 1 )(3a - 1) = (3a - 1)²


6x² - 7x + 2 = (3x - 2)(2x - 1)


2x² + 13x - 7 = (2x - 1)(x + 7)

Helen asks…

How do I factor trinomials with a lead coefficient larger than 1?

Here is the problem:
3x^4 + 5x^3 - 2x^2 = 0
x^2(3x^2+5x-2) = 0
x^2 (3x?__)(x?__) = 0

If you give me the answer, PLEASE show me the work first because the answer won't help me unless I can teach myself how to do it. Can anyone out there teach me how to do this exactly?

Thanks!

admin answers:

3x² + 5x - 2

One method is:

Multiply the first-term coefficient and the last, constant term including the minus sign. (3*-2 = -6).

Figure out two numbers that would multiply to this product (-6) and add up to the middle-term coefficient (+5). They are -1 and 6.

Break up the middle term into two parts, with these two numbers as the coefficients, in either order.
3x² + 6x - 1x - 2

Factor by grouping, breaking up the rewritten polynomial into two halves of two terms each and factoring each half by common factor.
3x(x + 2) - 1(x + 2)

Factor out the common binomial factor.
(3x - 1)(x + 2)

Steven asks…

How do you factor trinomials as the square of a binomial in algebra1 please help?

some examples from my homework are:
t^2-18t+81
25+10t + t^2
100a^2 - 20ab + b^2
49a^2 + 14a + 1
Please show work so i will have an example to do the others please help and thanks!

admin answers:

Unfortunately there is little work you can show for factoring. Basically you narrow down the list of possible factors and then use FOIL to check which ones are valid. Sort of like finding prime factors of integers.

1: t^2-18t+81 = (t - 9) (t - 9)
2: t^2+10t+25 = (t + 5) (t + 5)
3: 100a^2-20a+1 = ( 10a - b) ( 10a - b)
4: 49a^2+14a+1 = ( 7a + 1) ( 7a + 1)

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