# 3.1 – Factoring $x^2+bx+c$

## Objectives

• Define and write the general form of a quadratic trinomial.
• Find the binomial factors of a quadratic trinomial with a leading coefficient of 1.
• Determine the signs of a trinomial’s factors based on the sign of its x-term and its constant.

## Key Terms

• Binomial – A polynomial with exactly two terms.
• Constant Term – The term in a polynomial that does not have a variable. The constant term has a degree of zero.
• Degree – The value of the greatest exponent in a polynomial.
• ex. $7x^4-x^3+5x-9$ is a 4th degree polynomial
• Factor – To write a number or expression as a product of two or more numbers or expressions. A number or expression that is multiplied by another number or expression.
• The factors of a quadratic trinomial are two binomials
• FOIL – A method of multiplying binomials. FOIL stands for First, Outer, Inner, Last, the order in which you would multiply a binomial’s terms.
• Leading Coefficient – The number multiplied by the variable in the first term in a polynomial.
• Parabola – The set of points in a plane that are related to a given point (the focus) and a given line (the directrix) by this relationship: the distance from any point on the parabola to the focus is the same as the distance from that point to the directrix.
• A parabola is a conic section
• Polynomial – An algebraic expression with one or more terms. None of the variables is in the denominator of a fraction, and any exponents are whole numbers.
• Quadratic Function – A function that has a degree of 2.
• Quadratic Trinomial – A polynomial that has exactly three terms and a degree of 2.
• Trinomial – A polynomial that has exactly three terms.

## Notes

• Quadratic functions come in one of the following forms
• $y=ax^2+bx+c$
• $x=ay^2+by+c$

• Examples of quadratic trinomials
• $y=x^2+4y+1$
• $y=3x^2+4x-11$
• $y=6x^2-2y-8$
• Factoring Review: Binomials
• Example: 2x + 16
• Step 1: Decide which integer can be divided out of each term
• 2x and 16 are both divisible by 2
• Step 2: Factor out (reverse distribution) the 2
• 2 (x + 8) is the factorization

• Factoring Trinomials
• Example: $x^2$ is a trinomial if you write it like this: $x^2+0x+0$
• Answer: $(x + 0)(x + 0)$
Rules for Multiplying Binomials
Signs and Factors
Rules for Factoring