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Algebra tiles can be used to model polynomials.
These 1-by-1 square tiles have an area of
1 square unit.
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.
+ –+ – + –
1 –1 x –x x 2 –x 2
MODELING ADDITION OF POLYNOMIALS
You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1.
+ +
––
MODELING ADDITION OF POLYNOMIALS
+ + ++
+
+ + – –
1 Form the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1 with algebra tiles.
x 2 + 4x + 2
2 x 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.
+ + + + ++
+ ––+ + – –
x 2 + 4x + 2 2x
2 – 3x – 1
2 To add the polynomials, combine like terms. Group the x 2-tiles, the x-tiles, and the 1-tiles.
+
+ +
+
+
+ + + +
– – –
+
+ –=
MODELING ADDITION OF POLYNOMIALS
You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1.
+ + + + ++
+ ––+ + – –
x 2 + 4x + 2 2x
2 – 3x – 1
2 To add the polynomials, combine like terms. Group the x 2-tiles, the x-tiles, and the 1-tiles.
+
+ +
+
+
+ + + +
– – –
+
+ –=
3 Find and remove the zero pairs.
The sum is 3x 2 + x + 1.
An expression which is the sum of terms of the form a x k where k is a nonnegative
integer is a polynomial. Polynomials are usually written in standard form.
Adding and Subtracting Polynomials
Standard form means that the terms of the polynomial are placed in descending order, from largest degree to smallest degree.
The degree of each term of a polynomial is the exponent of the variable.
Polynomial in standard form:
2 x 3 + 5x 2 – 4 x + 7
Degree Constant termLeading coefficient
The degree of a polynomial is the largest degree of its terms. When a polynomial is written in standard form, the coefficient of the first term is the leading coefficient.
A polynomial with only one term is called a monomial. A polynomial with two terms
is called a binomial. A polynomial with three terms is called a trinomial. Identify
the following polynomials:
Classifying Polynomials
Polynomial DegreeClassified by
degreeClassified by
number of terms
6
–2 x
3x + 1
–x 2 + 2 x – 5
4x 3 – 8x
2 x 4 – 7x 3 – 5x + 1
0
1
1
4
2
3
constant
linear
linear
quartic
quadratic
cubic
monomial
monomial
binomial
polynomial
trinomial
binomial
Find the sum. Write the answer in standard format.
(5x 3 – x + 2 x 2 + 7) + (3x 2 + 7 – 4 x) + (4x 2 – 8 – x 3)
Adding Polynomials
SOLUTION
Vertical format: Write each expression in standard form. Align like terms.
5x 3 + 2 x 2 – x + 7
3x 2 – 4 x + 7
– x 3 + 4x 2 – 8+
4x 3 + 9x 2 – 5x + 6
Find the sum. Write the answer in standard format.
(2 x 2 + x – 5) + (x + x 2 + 6)
Adding Polynomials
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
Find the difference.
(–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4)
Subtracting Polynomials
SOLUTION
Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add.
–2 x 3 + 5x 2 – x + 8
–2 x 3 + 3x – 4– Add the opposite
No change –2 x 3 + 5x 2 – x + 8
2 x 3 – 3x + 4+
Find the difference.
(–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4)
Subtracting Polynomials
SOLUTION
Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add.
–2 x 3 + 5x 2 – x + 8
–2 x 3 + 3x – 4–
5x 2 – 4x + 12
–2 x 3 + 5x 2 – x + 8
2 x 3 – 3x + 4+
Find the difference.
(3x 2 – 5x + 3) – (2 x 2 – x – 4)
Subtracting Polynomials
SOLUTION
Use a horizontal format.
(3x 2 – 5x + 3) – (2 x 2 – x – 4) = (3x 2 – 5x + 3) + (–1)(2 x 2 – x – 4)
= x 2 – 4x + 7
= (3x 2 – 5x + 3) – 2 x 2 + x + 4
= (3x 2 – 2 x 2) + (– 5x + x) + (3 + 4)
Total Area = (10x)(14x – 2) (square inches)
Area of photo =
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.
Using Polynomials in Real Life
Write a model for the area of the mat around the photograph as a function of thescale factor.
Verbal Model
Labels
Area of mat = Area of photo
Area of mat = A
(5x)(7x)
(square inches)
(square inches)
Total Area –
Use a verbal model.
5x
7x
14x – 2
10x
SOLUTION
…
(10x)(14x – 2) – (5x)(7x)
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.
Using Polynomials in Real Life
Write a model for the area of the mat around the photograph as a function of thescale factor.
A =
= 140x 2 – 20x – 35x 2
SOLUTION
= 105x 2 – 20x
A model for the area of the mat around the photograph as a function of the scale factor x is A = 105x 2 – 20x.
AlgebraicModel
…
5x
7x
14x – 2
10x
