Sum Product Factoring

Sum Product Factoring is a commonly used method of factoring on trinomials \( (ax^2 + bx + c)\) where the value of \(a\) is usually 1. In order to factor the trinomial properly, we need to find 2 integers whose Product is \(c\) and Sum is \(b\). These integers (better known as factors) will be referred to as \(r\) and \(s\). Once these values are found, create a factored expression in the form \((x+r)(x+s)\).

Process for Factoring when \(a\) = 1

1. Identify the values of \(b\) and \(c\) in the expression which determine the sum and product resepctively
2. Create a table outlining the possible combinations of the sum and product based on the integer values
3. Arange the values in a factored expression n the form \((x+r)(x+s)\)

Example

Factor \(x^2+4x+3\).

First, we can identify that \(b = 4\) and \(c = 3\); as a result, we need to calculate 2 integers whose \(\text{sum} = 4\) and \(\text{product} = 3\).

Next, we can create a table that outlines the possible combinations for what the sum and product can be based on the \(2\) integer values. In this case, we try to identify what the sum is based on the product of \(3\):

r Value	1	-1
s Value	3	-3
Sum	4	-4
Product	3	3

Based on these results, we can determine that the \(r\) and \(s\) values (or factors) are \(1\) and \(3\) respectively.

Then, we can arrange the values in a factored expression:

\((x+1)(x+3)\)

Therefore, we can determine that the factored expression is \(\boldsymbol{(x+1)(x+3)}\).

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Process for Factoring when \(a\) ≠ 1

1. Identify the values of \(a\), \(b\), and \(c\) in the expression where \(b\) determines the sum and \(ac\) determines the product
2. Create a table outlining the possible combinations of the sum and product based on the integer values
3. Once the \(r\) and \(s\) values are determined, rewrite the trinomial as a 4-term expression: \(ax^2+rx+sx+c\)
4. Take the Greatest Common Factor (GCF) out of the 2 pairs of terms, \(ax^2+rx\) and \(sx+c\), to get common expressions
5. Simplify by placing the GCF terms in its own expression

Example

Factor \(2x^2+9x+10\).

First, we can identify that \(a = 2\), \(b = 9\) and \(c = 10\); as a result, we need to calculate \(2\) integers whose \(\text{product} = 20\) and \(\text{sum} = 9\).

Next, we can create a table that outlines the possible combinations for what the sum and product can be based on the \(2\) integer values. In this case, we try to identify what the sum is based on the product of \(20\):

r Value	1	-1	2	-2	4	-4
s Value	20	-20	10	-10	5	-5
Sum	21	-21	12	-12	9	-9
Product	20	20	20	20	20	20

Based on these results, we can determine that the \(r\) and \(s\) values (or factors) are \(4\) and \(5\) respectively.

Then, we can arrange the values in a factored expression in the form \(ax^2+rx+sx+c\):

\(2x^2+4x+5x+10\)

After, we can take out the GCF out of the \(2\) pairs of terms to get common expressions:

\(2x(x+2)+5(x+2)\)

From looking at the factored expression, we can identify the GCF in the first expression was \(2x\), the GCF in the second expression was \(5\) and the common expression was \(x+2\).

Finally, we can simplify this expression by placing the GCF terms in its own expression:

\((2x+5)(x+2)\)

Therefore, we can determine that the factored expression is \(\boldsymbol{(2x+5)(x+2)}\).

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