SOLUTION: When factoring a polynomial in the form ax2 + bx + c, where a, b, and c are positive real numbers, should the signs in the binomials be both positive, negative, or one of each? â

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: When factoring a polynomial in the form ax2 + bx + c, where a, b, and c are positive real numbers, should the signs in the binomials be both positive, negative, or one of each? â      Log On


   

Question 1204277: When factoring a polynomial in the form ax2 + bx + c, where a, b, and c are positive real
numbers, should the signs in the binomials be both positive, negative, or one of each?
• Create an example to verify your claim.
I need help so I can study an example like this for an upcoming test. Please help me with this.
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

if a, b, and c+are positive real numbers, binomials will be both positive
x%5E2+%2B+8x+%2B+15+=+%28x+%2B+5%29%28x+%2B+3%29

When c is positive the signs on the binomials are either both positive or both negative. The sign on b determines which.
x%5E2+%2B+8x+%2B+15+=+%28x+%2B+5%29%28x+%2B+3%29
x%5E2+-+8x+%2B+15+=+%28x+-+5%29%28x+-+3%29
When c is negative the signs on the binomials are one positive and one negative. The sign on b determines which is larger.
x%5E2+%2B+2x+-+15+=+%28x+%2B+5%29%28x+-+3%29
x%5E2+-+2x+-+15+=+%28x+-+5%29%28x+%2B3%29
If all terms of the trinomial are positive, then all terms of the binomials will be positive.
If the last term of the trinomial is negative but the middle term and the first term are positive, then one term of the binomial will be negative and the other will be positive.

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Both signs should be positive.

Let's consider: a = 1, b = 5, c = 6

The task is to factor x^2+5x+6

We need to find two numbers that multiply to 6 and add to 5.

Through fairly quick trial-and-error, you should find the two numbers to be 2 and 3.
2+3 = 5
2*3 = 6

Therefore, x^2+5x+6 factors to (x+2)(x+3)

x^2+5x+6 = (x+2)(x+3) is a true equation for any real number x. This is known as an identity.

-----------------------

Another example:

Factor x^2+12x+20

Here are the ways to multiply to 20 using positive integers
1*20
2*10
4*5

We see that 2 and 10 add to 12, so we have found the numbers we're after.

x^2+12x+20 = (x+2)(x+10)

-----------------------

Another example:

Factor 8x^2+22x+15

Unlike the other examples, the coefficient 'a' is not 1.
This will mean if you want to use the trial-and-error approach to factor, you'll need to use the AC method.
In this case the first and last coefficients multiply to 8*15 = 120.
So you'll need to find two numbers that multiply to 120 and add to 22.

It would be fairly inefficient to go through all of this trial-and-error since the numbers are larger.

Instead, we can use the quadratic formula to find the roots.
Then use those roots to construct the factors.

a = 8, b = 22, c = 15
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-22%2B-sqrt%28%2822%29%5E2-4%288%29%2815%29%29%29%2F%282%288%29%29

x+=+%28-22%2B-sqrt%28484+-+480%29%29%2F%2816%29

x+=+%28-22%2B-sqrt%284%29%29%2F%2816%29

x+=+%28-22%2B-++2%29%2F%2816%29

x+=+%28-22%2B2%29%2F%2816%29 or x+=+%28-22-2%29%2F%2816%29

x+=+%28-20%29%2F%2816%29 or x+=+%28-24%29%2F%2816%29

x+=+-5%2F4 or x+=+-3%2F2

Side note: you do not have to show all of these steps for your homework. This is just to show the student how the expression evaluates.

The roots x = -5/4 and x = -3/2 are then used to construct the factors.
x = -5/4 or x = -3/2
4x = -5 or 2x = -3
4x+5 = 0 or 2x+3 = 0
(4x+5)(2x+3) = 0

Use the FOIL rule, distributive property, or the box method to confirm that (4x+5)(2x+3) expands and simplifies to 8x^2+22x+15.

-----------------------

Need more examples?

Pick any four integers you want for p, q, r, and s.
Let's say we go for
p = 10
q = 3
r = 5
s = 7

The template
(px+q)(rx+s)
becomes
(10x+3)(5x+7)

Expand that out using your favorite method to get a quadratic of the form ax^2+bx+c.

This will help generate quadratics that can be factored over the integers.
Have your friends/classmates follow this process so they can give you practice problems, and vice versa.

Here are some other resources for further practice
https://www.mathsisfun.com/algebra/quadratic-factoring-practice.html
and
https://tutorial.math.lamar.edu/Problems/Alg/Factoring.aspx
Focus only on problems 7-15 for the 2nd link.