• Are there any types of quadratic expressions that you can factor without using
an area model? If so, what do these quadratics look like and how can you
recognize them? Today your team will examine several different “special”
quadratic expressions and look for patterns that can be used to factor them.
4-45. SPECIAL QUADRATICS
• Factor your team’s expressions, if possible (write product on the
right side with expo). Look for similarities and differences among
the expressions and their corresponding factored forms. Be
prepared to share your results with the class. Then work as a
class to sort them into groups based on patterns. What makes
these quadratics “special”?
4.1.5 FACTORING
SPECIAL CASES
November 19, 2015
Objectives
• CO: SWBAT factor special cases.
• LO: SWBAT explain in writing how
to recognize when a quadratic is a
perfect square trinomial or a
difference of squares.
4-46. Which of the following quadratic expressions fit the patterns you found in problem
4-45? Name if it is a perfect square trinomial or a difference of squares. Draw a box and
diamond if not. Then factor all if possible.
a.
25x2 − 1
b.
x2 – 5x − 36
c.
x2 + 8x + 16
Perfect square
trinomial
(x + 4)2
f.
9x2 − 100
Difference of
squares
(3x − 10)(3x + 10)
Difference of
squares
(5x − 1)(5x + 1)
(x − 9)(x + 4)
d.
9x2 − 12x + 4
Perfect square
trinomial
(3x − 2)2
e.
9x2 + 4
9x2 + 0x + 4
not factorable
4-48. CLOSED SETS: POLYNOMIALS
• ​Recall that in problem 4-38 you investigated whether
integers are a closed set under different operations. Now
you will focus on polynomials.
a. Are polynomials a closed set under addition? Are
polynomials a closed set under subtraction? That is, if
you add or subtract two polynomials, will you always get
a polynomial as your answer? Focus particularly on
linear and quadratic expressions. Give examples.
Polynomials are closed under addition and subtraction because you
will always get a polynomial when adding or subtracting polynomials.
b. Explain how you know your answer is always true.
When you combine like terms, you will still end up with a polynomial
(even if it’s zero)
4-49. LEARNING LOG - Factoring Special Cases
• Describe how to factor a difference of squares and a
perfect square trinomial. Be sure to include an example
of each type written as a sum and as a product.