Factoring Quadratics When a = 1

The standard form of a quadratic is ax² + bx + c, where and b are coefficients of x² and x, respectively.  So, for a = 1 means that the leading coefficient is 1.  You will only see x², not something like 5x².  

Each situation lends itself to a different way of factoring.  For the situation where a = 1, we will be looking at the numbers whose product is c and whose sum is b.  This will be slow at first, and the signs of the factors are a little slippery, initially.  

Read through the notes, taking notes yourself.  Then, watch the video.  After you’ve got a good idea of how it works it will be time to practice.  Use the tabs below to navigate the notes, videos and practice problems.

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The first method of factoring, factoring out a constant or monomial, is always relevant and possible applicable.  In other words, never forget to try applying that method.  For example:

3x2 – 6x – 72

Here there is a common factor of 3.  If we divide that 3 out, we have the following.

3(x2 – 2x – 24)

In this section we will learn to factor quadratic expressions with a leading coefficient of 1.  That is a = 1, as in ax2 + bx + c.  In case you forgot, when we learned about basic function types we discussed quadratic functions.  Those are functions with a degree (largest exponent) of two.  The standard form of a quadratic has coefficients a, b, and c.  An example is x2 – 2x – 24, which is the trinomial we found when we factored our initial example.  Here a = 1, b = -2, and c = -24. 

If factorable, these quadratics are the product of two binomials.    Let’s review how the FOIL short-cut works.  Consider (x – 6)(x + 4). 

Remember, it is often the case that the Outer and Inner products are like terms, and can be combined.  The first two terms of each binomial makes x2, the last two terms make – 24.  The outside and inside, +4x and -6x respectively, combine to make -2x.

What we’re going to do factor x2 – 2x – 24 is ask ourselves what two numbers multiply to negative twenty-four and also add to negative 2.  Since that is negative six and positive four, those are our last terms in our binomials that are factors of the trinomial.

Let’s look at five separate examples, looking at the factors, the product, and the question you need to ask to factor the quadratic expressions.

To factor one of these quadratic expressions, use what you learned about the patterns of multiplying binomials.  If a = 1, the question is always,

What two numbers multiply to “c” and add to “b”?

The signs are helpful, too.  Remember, that if c is positive, the factors are the same sign.  If c is negative the signs are different.  Look at the table above to see this is true.

Further, if c is positive and b is positive, the factors are both positive.  If c is positive and b is negative, both signs are negative.  If c is negative, the larger number will have the same sign as b.

In order to factor a quadratic expression with the leading coefficient of 1 (a = 1), there must be a pair of numbers whose product is the constant, c, and whose sum is the coefficient of x, the b value.  If no such numbers exist, the expression is prime, non-factorable.

Example 1:  Factor x2 – 5x + 6

Here c is positive and b is negative, so our values will both be negative.  We are looking for a pair of numbers whose product is 6 and whose sum is -5.  The only numbers that multiply to six are 1 and 6, and 2 and 3.  Since six is positive, the numbers are the same sign.  The factors are (x – 3)(x – 2).  The order does not matter, as multiplication has the commutative property.  However, the factors are unique, there is not another set of factors.  For example, 6 and 1 will not work.  In order to get a +6 as the product, they must be the same sign.  If they’re the same sign, then the sum is 7 or – 7.  

Last bit of advice:  Look for the numbers whose product is c first, not whose sum is b first.  There are too many numbers that can add to a given number, but a limited number of integers whose product will be c.