Basic Concepts Involved in Factoring Trinomials

Need some foundational material first?

Here, you will practice the basic concepts involved in factoring trinomials of the form $\,x^2 + bx + c\,.$ These trinomials have an $\,x^2\,$ 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 $\,f\,,$ $\,g\,$ and $\,x\,$:

\begin{align} &\cssId{s14}{(x+f)(x+g)}\cr\cr &\qquad \cssId{s15}{= \overset{\text{First}}{\overbrace{\strut\ x^2\ }}} \cssId{s16}{+ \overset{\text{Outer}}{\overbrace{\strut\ gx\ }}} \cssId{s17}{+ \overset{\text{Inner}}{\overbrace{\strut\ fx\ }}} \cssId{s18}{+ \overset{\text{Last}}{\overbrace{\strut\ fg\ }}}\cr\cr &\qquad \cssId{s19}{= x^2 + (f+g)x + fg} \end{align}

Now, think of going ‘backwards’:  from $\,x^2 + (f+g)x + fg\,$ back to the factored form $\,(x+f)(x+g)\,.$

We'd 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, which is the primary tool used in factoring trinomials:

KEY TOOL FOR FACTORING TRINOMIALS (where the coefficient of the squared term is $\,1\,$)
To factor a trinomial of the form $\,x^2 + bx + c\,,$ start by finding two numbers, $\,f\,$ and $\,g\,,$ that
• add together to give $\,b\,$ (the coefficient of the $\,x\,$ term); and
• multiply together to give $\,c\,$ (the constant term).
Then: \begin{align} &\cssId{s36}{x^2 + bx + c}\cr &\qquad \cssId{s37}{=\ \ x^2} \ \cssId{s38}{+ (\overset{= b}{\overbrace{f+g}})x} \cssId{s39}{+ \overset{= c}{\overbrace{\ fg\ }}}\cr\cr &\qquad \cssId{s40}{=\ \ (x + f)(x + g)} \end{align}

For example, to factor $\,x^2 + 5x + 6\,,$ we must find two numbers that add to $\,5\,$ and multiply to $\,6\,.$ The numbers $\,2\,$ and $\,3\,$ work, since $\,2+3 = 5\,$ and $2\cdot 3 = 6\,.$ Thus:

\begin{align} &\cssId{s48}{x^2 + 5x + 6}\cr &\qquad \cssId{s49}{=\ \ x^2 + (2 + 3)x + (2\cdot 3)}\cr &\qquad \cssId{s50}{=\ \ (x+2)(x+3)} \end{align}

(FOIL it out to check!)

When everything in sight is positive and coefficients are small, then it may be easy to come up with the ‘numbers that work’. For example, it may not be too hard for you to find numbers that add to $\,5\,$ and multiply to $\,6\,.$

However, bring some negative numbers into the picture and make coefficients bigger, and things can get considerably trickier.

Fortunately, there are some key ideas that will help you find the ‘numbers that work’ (if they exist), and the purpose of this web exercise is to give you practice with these ideas.

Key Ideas for Finding the ‘Numbers that Work’

• 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.

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.

Think of the number line interpretation of this fact. Start at zero. You must walk in only one direction (both numbers have the same sign). Each time you walk, you're getting farther from zero. Your final distance from zero is the sum of the individual distances you walk.

• 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.

Think of the number line interpretation of this fact. Start at zero. You must walk in both directions (the numbers have different signs). So, you'll be doing some back-tracking (some overlapping). Your final distance from zero is the difference of the individual distances you walk.

• 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.

Think of the number line interpretation of this fact. Start at zero. You need to walk in only one direction (both numbers have the same sign). You need to end up to the right of zero (the sum is positive). So, you must walk to the right both times (both numbers must 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.

Think of the number line interpretation of this fact. Start at zero. You need to walk in only one direction (both numbers have the same sign). You need to end up to the left of zero (the sum is negative). So, you must walk to the left both times (both numbers must 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.)

Think of the number line interpretation of this fact. Start at zero. You must walk in both directions (the numbers have different signs). You need to end up to the right of zero (the sum is positive). So, you must walk farther to the right (the bigger number must be positive).

• 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.)

Think of the number line interpretation of this fact. Start at zero. You must walk in both directions (the numbers have different signs). You need to end up to the left of zero (the sum is negative). So, you must walk farther to the left (the bigger number must be negative).

Examples

Question: Suppose two numbers multiply to $\,36\,.$ Then, the numbers have (choose one):
 THE SAME SIGN DIFFERENT SIGNS
Question: Suppose two numbers multiply to $\,-36\,.$ Then, the numbers have (choose one):
 THE SAME SIGN DIFFERENT SIGNS
Question: Suppose two numbers have the same sign, and they add to $\,10\,.$ Then, the numbers are (choose one):
 BOTH POSITIVE BOTH NEGATIVE
Question: Suppose two numbers have the same sign, and they add to $\,-10\,.$ Then, the numbers are (choose one):
 BOTH POSITIVE BOTH NEGATIVE
Question: When you add two numbers that have the same sign, then in your head you do a/an (choose one):
Question: Suppose two numbers have different signs, and they add to $\,10\,.$ Then, the bigger number is (choose one):
Question: Suppose two numbers have different signs, and they add to $\,-10\,.$ Then, the bigger number is (choose one):