Basic Concepts Involved in Factoring Trinomials

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BASIC CONCEPTS INVOLVED IN FACTORING TRINOMIALS

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Here, you will practice the basic concepts involved in factoring trinomials of the form x2 +cx+d .
These trinomials have an x2 term with a coefficient of 1 , an x term, and a constant term.

Recall that factoring is the process of taking a sum (things added)
and rewriting it as a product (things multiplied).

Observe that for all real numbers a , b , and x ,
(x+a) (x+b)   =  x2 +b⁢x+ a⁢x+a ⁢b  =  x 2+( a+b)x+ a⁢b .

Now, think of going from x 2+( a+b)x+ a⁢b
back to the factored form (x+a) (x+b) .

You need two numbers that ADD TOGETHER to give the coefficient of the x term,
and that MULTIPLY TOGETHER to give the constant term.

This gives the following result:

KEY RESULT FOR FACTORING TRINOMIALS
(where the coefficient of the squared term is 1 )

To factor a trinomial of the form x2 +cx+d ,
start by finding two numbers, a and b , that:

ADD TOGETHER to give c (the coefficient of the x term); and
MULTIPLY TOGETHER to give d (the constant term).

Then, x2 +c⁢x +d=(x+ a)(x+b ) .

For example, to factor x2 +5x+6 ,
we must find two numbers that add to 5 and multiply to 6 .
The numbers 2 and 3 work,
since 2+3=5 and 2⋅3= 6 .
Thus, x2 +5x+6 =(x+2) (x+3) .
(FOIL it out to check!)

There are several key ideas that come up in finding the "numbers that work,"
and the purpose of this web exercise is to give you practice with these central ideas.

If two numbers multiply to give a POSITIVE number, then they must both be positive or they must both be negative.
That is, two numbers that multiply to a positive number must have the same sign.

If two numbers multiply to give a NEGATIVE number, then one must be positive, and the other must be negative.
That is, two numbers that multiply to a negative number must have different signs.

When you add two numbers that have the same sign, then in your head you do an addition problem.
For example, to mentally compute the sum (-2) + (-5) , in your head you would compute 2 + 5 , and then assign a negative sign to your answer.

When you add two numbers that have different signs, then in your head you do a subtraction problem.
For example, to mentally compute -5 + 2 , you would think 5 - 2 , and then make the answer negative.

If two numbers have the same sign and their sum is positive, then both numbers must be positive.
For example, if two numbers have the same sign and add to 10 , then they must both be positive.

If two numbers have the same sign and their sum is negative, then both numbers must be negative.
For example, if two numbers have the same sign and add to -10 , then they must both be negative.

If two numbers have different signs and their sum is positive, then the bigger number must be positive.
(Remember that "bigger" means farther away from zero.)
For example, if two numbers have different signs and add to 10 , then the bigger number must be positive.
(Like 12 + (-2) = 10 : the numbers being added are 12 and -2 ; 12 is bigger because it is farther from zero.)

If two numbers have different signs and their sum is negative, then the bigger number must be negative.
(Remember that "bigger" means farther away from zero.)
For example, if two numbers have different signs and add to -10 , then the bigger number must be negative.
(Like -12 + 2 = -10 : the numbers being added are -12 and 2 ; -12 is bigger because it is farther from zero.)

EXAMPLES:

Suppose two numbers multiply to 36.
Then, the numbers have (choose one):     THE SAME SIGN      DIFFERENT SIGNS  .
Answer:   THE SAME SIGN

Suppose two numbers multiply to -36.
Then, the numbers have (choose one):     THE SAME SIGN      DIFFERENT SIGNS  .
Answer:   DIFFERENT SIGNS

Suppose two numbers have the same sign, and they add to 10.
Then, the numbers are (choose one):     BOTH POSITIVE     BOTH NEGATIVE  .
Answer:   BOTH POSITIVE

Suppose two numbers have the same sign, and they add to -10.
Then, the numbers are (choose one):     BOTH POSITIVE     BOTH NEGATIVE  .
Answer:   BOTH NEGATIVE

When you add two numbers that have the same sign,
then in your head you do a (choose one):     ADDITION PROBLEM     SUBTRACTION PROBLEM  .
Answer:   ADDITION PROBLEM

When you add two numbers that have different signs,
then in your head you do a (choose one):     ADDITION PROBLEM     SUBTRACTION PROBLEM  .
Answer:   SUBTRACTION PROBLEM

Suppose two numbers have different signs, and they add to 10.
Then, the bigger number is (choose one):     POSITIVE    NEGATIVE  .
Answer:   POSITIVE

Suppose two numbers have different signs, and they add to -10.
Then, the bigger number is (choose one):     POSITIVE    NEGATIVE  .
Answer:   NEGATIVE

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.