# Week 5 – Precalc 11

This week in Precalculus 11, we started the Solving Quadratic Equations Unit. We first reviewed factoring polynomial expressions.

Factoring: separating an expression into its components

Polynomial expression: an expression of numbers and variables being added, subtracted or multiplied (example: 2x)

Greatest common factor: the greatest number that can divide into all the terms in the expression (example: 2, 4, 6, 8, 10      GCF = 2)

Term: a number or a group of numbers being multiplied (example: 2$x^2$)

Binomial: a polynomial expression with 2 terms (example: x + 1)

Difference of squares: a polynomial expression in which subtraction takes place between 2 perfect square terms (example: 4x – 1)

Trinomial: a polynomial expression with 3 terms (example: $x^2$ + x + 1)

Conjugates: 2 terms with opposite addition/subtraction signs (example: 1 + 1 & 1 – 1)

To factor a polynomial expression, we first look for the greatest common factor between the terms and divide each term by that number.

Example:

2x + 2

= 2(x + 1)

If the expression is a binomial, we check if it is a difference of squares. When factored, a difference of squares results in conjugates.

Example:

4x – 1

= (2x + 1)(2x – 1)

If the expression is a trinomial, we check if it is in the form a$x^2$ + bx + c. If a = 1, we separate bx into 2 terms that multiply to c and add to bx.

Example:

$x^2$ + 2x + 1

= (x + 1)(x + 1)

= $(x + 1)^2$

If a ≠ 1, we separate bx into 2 terms that multiply to ac and add to bx, then find the greatest common factor of each side, and divide each term by that number.

Example:

2$x^2$ + 4x + 2

= 2$x^2$ + 2x + 2x + 2

= 2x(x + 1) + 2(x + 1)

= (2x + 2)(x + 1)

If the expression is in a different form than a$x^2$ + bx + c, the expression can sometimes be changed to this form.

Example:

$(x + 1)^2$ + 2(x + 1) + 1

= $a^2$ 2a + 1

= (a + 1)(a + 1)

= $(a + 1)^2$

= $[(x + 1) + 1]^2$

= $(x + 2)^2$