Transcript Chapter 8-5: Adding and Subtracting Polynomials

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 and Subtracting Polynomials
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.
Standard form means that the terms of the polynomial are placed in descending
order, from largest degree to smallest degree.
Polynomial in standard form:
2 x 3 + 5x 2 – 4 x + 7
Leading coefficient
Degree
Constant term
The degree of each term of a polynomial is the exponent of the variable.
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.
Classifying Polynomials
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:
Degree
Classified by
degree
Classified by
number of terms
6
0
constant
monomial
–2 x
1
linear
monomial
3x + 1
1
linear
binomial
–x 2 + 2 x – 5
2
quadratic
trinomial
4x 3 – 8x
3
cubic
binomial
2 x 4 – 7x 3 – 5x + 1
4
quartic
polynomial
Polynomial
Adding Polynomials
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)
SOLUTION
Vertical format: Write each expression in standard form. Align like terms.
5x 3 + 2 x 2 – x + 7
3x 2 – 4 x + 7
3
2
+ – x + 4x
–8
4x 3 + 9x 2 – 5x + 6
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
Subtracting Polynomials
Find the difference.
(–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4)
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
No change
Add the opposite
–2 x 3 + 5x 2 – x + 8
+ 2 x3
– 3x + 4
Subtracting Polynomials
Find the difference.
(–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4)
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
–2 x 3 + 5x 2 – x + 8
+ 2 x3
– 3x + 4
5x 2 – 4x + 12
Subtracting Polynomials
Find the difference.
(3x 2 – 5x + 3) – (2 x 2 – x – 4)
SOLUTION
Use a horizontal format.
(3x 2 – 5x + 3) – (2 x 2 – x – 4) = (3x 2 – 5x + 3) + (–1)(2 x 2 – x – 4)
= (3x 2 – 5x + 3) – 2 x 2 + x + 4
= (3x 2 – 2 x 2) + (– 5x + x) + (3 + 4)
= x 2 – 4x + 7
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.