A.SSE.2: Factor Polynomials
A.SSE.2: Factor Polynomials
Interpret the structure of expressions.
2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it
as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
Overview of Lesson
- activate prior knowledge
- present vocabulary and/or big ideas associated with the lesson
- connect assessment practices with curriculum
- model an assessment problem and solution strategy
- facilitate guided practice of student activity
Optional: Provide or allow students to create additional problem sets
- facilitate a summary and share out of student work
Optional HW - Write the math assignment.
Big Ideas
Factoring polynomials is one of four general methods taught in the Regents mathematics curriculum for
finding the roots of a quadratic equation. The other three methods are the quadratic formula, completing
the square and graphing.
The roots of a quadratic equation can found using the factoring method when the discriminant’s
value is equal to either zero or a perfect square.
Using the Discriminant to Predict Types of Roots
x-axis
Two Real Roots: Parabola
intersects the x-axis in two
places.
x-axis
One Real Root: Parabola
touches the x-axis in one
place.
x-axis
Imaginary Roots: Parabola
does not intersect the x-axis.
Prime Factoring of a Monomial:
204 x2  2 102 x2  2  2 51x2  2  2  3 17 x 2  22  3 17  x 2






Factoring Binomials: NOTE: This is the inverse of the distributive property.
3  x  2   3x  6
2 x 2  6 x  2 x  x  3
Special Case: Factoring the Difference of Perfect Squares.

General Rule
2
a  b2   a  b  a  b 

Examples
x 2  4   x  2  x  2 
x 4  9   x 2  3 x 2  3
Special Case: Factoring the Sum and Difference of Perfect Cubes.
General Rule
Examples
3
3
2
2
3
a  b  (a  b)  a - ab  b 
x  27  ( x  3)  x 2 - 3 x  32 
a 3  b3  (a - b)  a 2  ab  b 2 
x3  125  ( x - 5)  x 2  5 x  52 
How to Factor Trinomials.
Given a trinomial in the form ax 2  bx  c  0 whose discriminant equals zero or a perfect square, it may
be factored as follows:
STEP 1. The product of these two numbers
must equal c.
ax 2  bx  c  0 

x ___

x ___

STEP 2. The signs of these two numbers are
determined by the signs of b and c.
Inners
ax 2  bx  c  0 

x ___

x ___

Outers
STEP 3. The product of the outer numbers plus the product of the inner numbers
must sum to b.
Modeling:
x2  5x  6   x  2 x  3
2x2  8x  6   2x  2 x  3
4x2 10x  6   2x  2 2x  3
Turning Factors into Roots. Students frequently do not understand why each factor of a binomial or
trinomial can be set to equal zero, thus leading to the roots of the equation. Recall that the standard form
of a quadratic equation is ax 2  bx  c  0 and only the left side of the equation is factored. Thus, the
left side of the equation equals zero.
For all numbers
a 0  0
and if
a b  0
b0
(NOTE: substitute any two factors)
Therefore
a 0
REGENTS PROBLEMS
1. Which expression is equivalent to
a.
b.
?
c.
d.
2. When factored completely, the expression
a.
c.
is equivalent to
b.
d.
3. Four expressions are shown below.
I
II
III
IV
The expression
a. I and II, only
b. II and IV, only
is equivalent to
c. I, II, and IV
d. II, III, and IV
4. If the area of a rectangle is expressed as
could be expressed as
a.
b.
5. Factor the expression
6. The zeros of the function
a.
and
b. 1 and
, then the product of the length and the width of the rectangle
c.
d.
completely.
are
c. 1 and 2
d.
and 2
7. If Lylah completes the square for
in order to find the minimum, she must write
general form
. What is the value of a for
?
a. 6
c. 12
b.
d.
in the
A.SSE.2: Factor Polynomials
Answer Section
1. ANS: A
Strategy 1. Factor
Strategy 2. Work backwards using the distributive property to check each answer choice.
a
c
(correct)
b
(wrong)
d
(wrong)
(wrong)
PTS: 2
REF: 081415a1
NAT: A.SSE.2
2. ANS: C
Strategy: Use difference of perfect squares.
STEP 1. Factor
STEP 2. Factor
TOP: Factoring Polynomials
PTS: 2
REF: 011522a1
NAT: A.SSE.2
TOP: Factoring Polynomials
3. ANS: C
Strategy: Use the distributive property to expand each expression, then match the expanded expressions to the
answer choices.
I
III
II
IV
Answer choice c is correct.
PTS: 2
REF: 081509ai
NAT: A.SSE.2
TOP: Factoring Polynomials
4. ANS: B
Strategy: Use the distributive property to work backwards from the answer choices.
a.
c.
b.
PTS: 2
5. ANS:
d.
REF: 061503AI
NAT: A.SSE.2
Strategy: Factor the trinomial, then factor the perfect square.
STEP 1. Factor the trinomial
.
TOP: Factoring Polynomials
STEP 2. Factor the perfect square.
PTS: 2
REF: 061431a1
NAT: A.SSE.2
TOP: Factoring Polynomials
6. ANS: D
Strategy 1. Factor, then use the multiplication property of zero to find zeros.
Strategy 2. Use the quadratic formula.
Strategy 3. Input into graphing calculator and inspect table and graph.
PTS: 2
REF: 081513ai
7. ANS: A
Strategy: Transform
If
, then
NAT: A.SSE.3
into the form of
TOP: Solving Quadratics
and find the value of a.
.
PTS: 2
REF: 081520ai
KEY: completing the square
NAT: A.SSE.3
TOP: Solving Quadratics