Factoring trinomials is the process of rewriting a quadratic expression as a product of two binomials. This matters because factored form often makes equations easier to solve, graph, and interpret. A trinomial such as x^2 + bx + c can often be broken into (x + m)(x + n), where m and n are carefully chosen numbers.
Learning the pattern helps students connect multiplication, area models, and quadratic equations.
Key Facts
- A trinomial has three terms, such as ax^2 + bx + c.
- For x^2 + bx + c, find two numbers m and n with m + n = b and mn = c.
- If x^2 + bx + c = (x + m)(x + n), then b = m + n and c = mn.
- For ax^2 + bx + c, the AC method uses two numbers that multiply to ac and add to b.
- After splitting the middle term, factor by grouping: ax^2 + px + qx + c = x(ax + p) + r(ax + p).
- Always check by expanding: (mx + n)(px + q) = mpx^2 + (mq + np)x + nq.
Vocabulary
- Trinomial
- A polynomial with exactly three terms, such as 2x^2 + 7x + 3.
- Quadratic expression
- An expression whose highest power of the variable is 2.
- Binomial factor
- A two-term expression that multiplies with another factor to make the original expression.
- AC method
- A factoring method for ax^2 + bx + c that uses the product ac and the sum b to split the middle term.
- Factoring by grouping
- A method that groups terms in pairs so a common binomial factor can be pulled out.
Common Mistakes to Avoid
- Using numbers that multiply to b instead of c for x^2 + bx + c is wrong because the constant term comes from multiplying the two constants in the binomials.
- Ignoring the sign of c is wrong because a negative c means the two chosen numbers must have opposite signs.
- Factoring ax^2 + bx + c as if a = 1 is wrong when a is not 1 because the leading coefficient changes the possible binomial factors.
- Forgetting to check by expanding is risky because a small sign error can produce factors that look reasonable but do not equal the original trinomial.
Practice Questions
- 1 Factor x^2 + 9x + 20 completely.
- 2 Use the AC method to factor 6x^2 + 11x + 3 completely.
- 3 Explain why x^2 - 5x + 6 factors using two negative numbers, and describe how the signs of b and c tell you this.