Sign patterns for factoring
Sign Patterns for Factoring provides Grade 6 students with rules to determine the signs of the constants p and q when factoring trinomials of the form x² + bx + c into (x + p)(x + q). Covered in Yoshiwara Elementary Algebra Chapter 6: Quadratic Equations, the sign patterns are: if c is positive, p and q have the same sign as b; if c is negative, p and q have opposite signs with the larger absolute value matching the sign of b. Recognizing these patterns speeds up factoring significantly.
Key Concepts
Property Assume that $b$, $c$, $p$, and $q$ are positive integers. 1. $x^2 + bx + c = (x + p)(x + q)$. If all the coefficients of the trinomial are positive, then both $p$ and $q$ are positive. 2. $x^2 bx + c = (x p)(x q)$. If the linear term of the trinomial is negative and the other two terms positive, then $p$ and $q$ are both negative. 3. $x^2 \pm bx c = (x + p)(x q)$. If the constant term of the trinomial is negative, then $p$ and $q$ have opposite signs.
Examples To factor $x^2 7x + 10$, the constant term is positive and the middle term is negative. So, we need two negative numbers that multiply to 10 and add to 7. The factors are $(x 2)(x 5)$. In $y^2 + 4y 12$, the constant term is negative, so the factors have opposite signs. We need numbers that multiply to 12 and add to 4. The factorization is $(y + 6)(y 2)$. For $z^2 z 30$, the negative constant term means opposite signs. We need numbers that multiply to 30 and add to 1. The correct pair is 6 and 5, so we get $(z 6)(z + 5)$.
Explanation The signs in the trinomial give you huge clues! A positive last term means the signs in your factors are the same. A negative last term means the signs are different. Use this to factor faster!
Common Questions
What are the sign patterns for factoring trinomials?
If c > 0 and b > 0, both factors are positive. If c > 0 and b < 0, both factors are negative. If c < 0, the factors have opposite signs, with the larger absolute value matching the sign of b.
How do sign patterns help factor x² - 5x + 6?
Since c = 6 > 0 and b = -5 < 0, both constants are negative. Find two negative numbers that multiply to 6 and add to -5: -2 and -3. So it factors as (x - 2)(x - 3).
What if c is negative in the trinomial?
When c is negative, the two constants in the factored form have opposite signs. The one with the larger absolute value takes the sign of the middle coefficient b.
Where are sign patterns for factoring in Yoshiwara Elementary Algebra?
Sign patterns are covered in Chapter 6: Quadratic Equations of Yoshiwara Elementary Algebra.
How do you use sign patterns with trial and error?
Start by determining the signs required, then only test factor pairs with those signs. This eliminates half or more of the possibilities.