Sept. 26 Class

Learning Goals: Understand how to factor using perfect squares and difference of squares.

Today we did the formative quiz and took it up in class.

Formative Quiz

Formative Quiz Solutions

Then we discussed some special cases of factoring.

Perfect square trinomials are ones that satisfy this pattern:

a²x² ± 2abx + b²

These types of trinomials can be factored into: (ax ± b)².

Example  Factor x² – 16x + 64.

Solution We notice that the first and last terms are perfect squares.  We check if it fits the pattern.

a = 1, b = 8, 2ab = 16.   This matches the pattern, so the solution is,

(x – 8)²

We could have found the solution using other methods, but this way is much faster.

Example  Factor 4x² + 20x + 25.

Solution We notice that the first and last terms are perfect squares.  We check if it fits the pattern.

a = 2, b = 5, 2ab = 20.   This matches the pattern, so the solution is,

(2x + 5)²

Difference of Squares refers to an expression with two perfect squares and a minus sign between the two.

a² – b²

In this case the factors are,

(a – b)(a+b)

Example Factor 4x² + 49.

Solution This fits the pattern, so the solution is (2x – 7)(2x + 7)

Example Factor 3x² – 27.

Solution Always common factor first if possible, 3(x² – 9)

Now there is a perfect square in the brackets.  3(x – 3)(x+3)

Example Factor 4x² – (3x + 1)².

Solution (3x + 1)² is a perfect square!      [2x – (3x + 1)][2x + (3x + 1)]

This simplifies to -(x + 1)(5x + 1).

Homework: p. 115 #2-12