# 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 $$x^2+4x+3$$

1. We identify that $$b$$ = 4 and $$c$$ = 3; as a result, we need to calculate 2 integers whose $$\text{sum} = 4$$ and $$\text{product} = 3$$.

2. 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 s value 1 -1 3 -3 4 -4 3 3

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

3. We can arrange the values in a factored expression, $$(x+1)(x+3)$$.

Factor $$x^2+7x-18$$

Enter in coefficients for the trinomial or click on the button to generate a new question. Then, factor using what you've learnt.

## 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 $$2x^2+9x+10$$

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

2. 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 s value Sum Product 1 -1 2 -2 4 -4 20 -20 10 -10 5 -5 21 -21 12 -12 9 -9 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.

3. We can arrange the values in a factored expression in the form $$ax^2+rx+sx+c$$ which in this case will be $$2x^2+4x+5x+10$$

4. We can take out the GCF out of the 2 pairs of terms to get common expressions: $$2x(x+2)+5(x+2)$$

5. The GCF in the first expression was $$2x$$, the GCF in the second expression was 5 and the common expression was $$x+2$$

6. Simplify by placing the GCF terms in its own expression: $$(2x+5)(x+2)$$

Factor $$-3x^2-4x+20$$