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)
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 + 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.
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.
4x – 1
= (2x + 1)(2x – 1)
If the expression is a trinomial, we check if it is in the form a + bx + c. If a = 1, we separate bx into 2 terms that multiply to c and add to bx.
+ 2x + 1
= (x + 1)(x + 1)
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.
2 + 4x + 2
= 2 + 2x + 2x + 2
= 2x(x + 1) + 2(x + 1)
= (2x + 2)(x + 1)
If the expression is in a different form than a + bx + c, the expression can sometimes be changed to this form.
+ 2(x + 1) + 1
= 2a + 1
= (a + 1)(a + 1)