Factoring by guess-and-check
Factoring by Guess-and-Check teaches Grade 6 students to factor quadratic trinomials ax² + bx + c by working backwards from FOIL — testing combinations of first and last terms for two binomials until the middle term checks out. Covered in Yoshiwara Elementary Algebra Chapter 7: Polynomials, this method develops pattern recognition and algebraic reasoning. The first terms of the binomials must multiply to ax², the last terms must multiply to c, and the cross-multiplication must give the bx middle term.
Key Concepts
Property To factor a quadratic trinomial of the form $ax^2 + bx + c$, we reverse the FOIL method. We look for two binomials whose product matches the trinomial.
The F irst terms of the binomials must multiply to $ax^2$. The L ast terms must multiply to $c$. The sum of the O uter and I nner products must equal the middle term, $bx$.
Examples To factor $2x^2 7x + 3$, the first terms could be $2x$ and $x$. The last terms could be $ 1$ and $ 3$. The combination $(2x 1)(x 3)$ gives a middle term of $ 6x x = 7x$, so it is correct.
Common Questions
How do you factor a trinomial by guess-and-check?
Look for two binomials (px + q)(rx + s) where pr = a (leading coefficient), qs = c (constant), and ps + qr = b (middle coefficient). Test possibilities until the middle term matches.
Where do you start when factoring ax² + bx + c?
List all factor pairs of a for the first terms and all factor pairs of c for the last terms. Then check which combination gives the correct middle term b.
How do you verify a factoring answer?
Use FOIL to multiply your factored binomials. If the result equals the original trinomial, the factoring is correct.
Where is factoring by guess-and-check in Yoshiwara Elementary Algebra?
It is in Chapter 7: Polynomials of Yoshiwara Elementary Algebra.
When should you use guess-and-check vs. other factoring methods?
Guess-and-check works well when coefficients are small. For larger coefficients, the AC method (factor by grouping) is often more systematic.