# 3.2 – $ax^2+bx+c$

## Objectives

• List the steps used to factor a trinomial that has a leading coefficient other than 1.
• Find the factors of a trinomial that has a leading coefficient that is not 1.
• Find the trinomial for a given set of factors.

## Key Terms

• Common Factor – A number or expression that is a factor of two or more numbers or polynomials.
• Factor It Out – To write an expression as the product of a factor and another expression. The factor must be common to each term of the original expression. To factor out a common factor from an expression is equivalent to using the distributive property in reverse.
• Factorization – The result of writing a number or expression as a product of two or more factors.
• Reducible Trinomial – A polynomial that has exactly three terms and is able to be factored.
• Any reducible trinomial can be factored as  $(rx\pm p)(sx\pm q)$, where r•s is the trinomial’s leading coefficient and p•q is the constant term

## Notes

Greatest Common Factor (GCF)
• Rule: When the leading coefficient is a number other than 1, the first step is to look for a common factor in each term and factor it out.

• Trinomials
• If you multiply the x-coefficients of the factors, you get the leading coefficient of the trinomial
• If you multiply the constant terms of the factors, you get the constant term of the trinomial
• The middle term depends on the factors and the signs.  You will need to use “trial and error” with the OI in FOIL.
• FOIL: First, Outer, Inner, Last
• FOIL is a form of distribution

• To factor  $ax^2 + bx + c$, you must find all possible values of r, s, p, and q
• Make two lists:
• List 1: r and s
• These are the factors of the leading coefficient
• List 2: p and q
• These are the factors of the constant term
Examples
•  Ex 1:  $8x^2+12x+4$

• Use Trial and Error methods to find the right solution(s)
• Try all the possibilities and see which ones work
• Both of the following factorizations work:
• First possibility: (4x + 4)(2x + 1)
• Factor out the 4 from the first binomial: 4(x + 4)
• 4(x + 1)(2x + 1)
• Second possibility: (2x + 2)(4x + 2)
• Factor out a 2 from each binomial: 2(x + 2) and 2(2x + 1)
• Since 2*2 = 4, put the 4 in the front: 4(x + 1)(2x + 1)

• Ex 2