Adding & Subtracting
Polynomials
A-APR.4 Objective:

Prove polynomial identities and use them
to describe numerical relationships. For
example, the polynomial identity (x2 + y2)2
= (x2 – y2)2 + (2xy)2 can be used to
generate Pythagorean triples.
MODELING ADDITION OF POLYNOMIALS
Algebra tiles can be used to model polynomials.
+
–
1
–1
These 1-by-1 square
tiles have an area of
1 square unit.
+
–
+
–
x
–x
x2
–x 2
These 1-by-x rectangular
tiles have an area of x
square units.
These x-by-x rectangular
tiles have an area of x 2
square units.
MODELING ADDITION OF POLYNOMIALS
You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1.
1
Form the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1 with algebra tiles.
x2
+
4x
+ 2
+
+
+
+
+
+
+
–
2 x2
+
+
– 1
3x
–
–
–
–
MODELING ADDITION OF POLYNOMIALS
You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1.
2
To add the polynomials, combine like terms. Group the x 2-tiles, the x-tiles,
and the 1-tiles.
x 2 + 4x + 2
+
+
+
+
+
+
+
+
2x 2 – 3x – 1
+
+
–
+
+
+
+
+
+
+
+
+
=
+
–
–
–
–
–
–
–
MODELING ADDITION OF POLYNOMIALS
You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1.
2
3
To add the polynomials, combine like terms. Group the x 2-tiles, the x-tiles,
and the 1-tiles.
2 + 4x + 2
xFind
and remove the zero pairs. +
+
+ sum is
+ 3x+2 + x+ + 1.
+
The
+
2x 2 – 3x – 1
+
+
–
+
+
+
+
+
+
+
+
+
=
+
–
–
–
–
–
–
–
Adding Polynomials
Find the sum. Write the answer in standard format.
(2 x 2 + x – 5) + (x + x 2 + 6)
SOLUTION
Horizontal format: Add like terms.
(2 x 2 + x – 5) + (x + x 2 + 6) = (2 x 2 + x 2) + (x + x) + (–5 + 6)
= 3x 2 + 2 x + 1
Using Polynomials in Real Life
You are enlarging a 5-inch by 7-inch photo by a scale factor of x and mounting it on
a mat. You want the mat to be twice as wide as the enlarged photo and 2 inches
less than twice as high as the enlarged photo.
Write a model for the area of the mat around the photograph as a function of the
scale factor.
Use a verbal model.
Verbal Model
Area of mat = Total Area –
Labels
…
Area of
photo
7x
Area of mat = A
(square inches)
5x
Total Area = (10x)(14x – 2)
(square inches)
10x
Area of photo = (5x)(7x)
(square inches)
14x – 2
SOLUTION
Using Polynomials in Real Life
You are enlarging a 5-inch by 7-inch photo by a scale factor of x and mounting it on
a mat. You want the mat to be twice as wide as the enlarged photo and 2 inches
less than twice as high as the enlarged photo.
Write a model for the area of the mat around the photograph as a function of the
scale factor.
SOLUTION
…
= 140x 2 – 20x – 35x 2
5x
= 105x 2 – 20x
10x
14x – 2
Algebraic
Model
7x
A = (10x)(14x – 2) – (5x)(7x)
A model for the area of the mat around the photograph as a function of the
scale factor x is A = 105x 2 – 20x.