Factoring Quadratic Expressions (Leading Coefficient 1)
Factoring quadratic expressions with a leading coefficient of 1 is a Grade 11 algebra skill in Big Ideas Math. For x² + bx + c, find two integers whose product equals c and whose sum equals b. Write the factored form as (x + p)(x + q) where p × q = c and p + q = b. For example, x² + 5x + 6: find two numbers with product 6 and sum 5—that's 2 and 3—giving (x + 2)(x + 3). For x² − x − 6: product −6, sum −1—that's −3 and 2—giving (x − 3)(x + 2). Factoring is confirmed by FOIL multiplication to verify the original expression is recovered.
Key Concepts
To factor a quadratic expression of the form $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$. The factored form is $(x + m)(x + n)$ where $m \cdot n = c$ and $m + n = b$.
Common Questions
How do you factor x² + bx + c when the leading coefficient is 1?
Find two integers p and q where p × q = c and p + q = b. Write the factored form as (x + p)(x + q).
How do you factor x² + 5x + 6?
Find two numbers with product 6 and sum 5: that's 2 and 3. Factor: (x + 2)(x + 3).
How do you factor x² − x − 6?
Find two numbers with product −6 and sum −1: that's −3 and 2. Factor: (x − 3)(x + 2).
How do you factor x² − 7x + 12?
Find two numbers with product 12 and sum −7: that's −3 and −4. Factor: (x − 3)(x − 4).
How do you verify that a factoring is correct?
Expand using FOIL: (x + p)(x + q) = x² + qx + px + pq = x² + (p+q)x + pq. Check that this matches the original expression.
What should you do if no integer pair has the required product and sum?
If no integer pair works, the expression may be prime over integers (cannot be factored with integer coefficients). Use the quadratic formula to find roots instead.