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Algebra tiles can be used to model polynomials.
These 1-by-1square
tiles have an area of
1square unit.
These 1-by-xrectangular
tiles have an area of x
square units.
These x-by-xrectangular
tiles have an area of x2
square units.
+ + +
1 1 x x x2 x2
MODELING ADDITION OF POLYNOMIALS
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You can use algebra tiles to add the polynomials x2+ 4x+ 2and 2x23x1.
+ +
MODELING ADDITION OF POLYNOMIALS
+ + + +
+
+ +
1 Form the polynomials x2+ 4x+ 2and 2x23x1with algebra tiles.
x2 + 4x + 2
2x2 3x 1
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MODELING ADDITION OF POLYNOMIALSYou can use algebra tiles to add the polynomials x2+ 4x+ 2and 2x23x1.
+ + + + + +
+
+ +
x2+ 4x+ 2 2x23x1
2 To add the polynomials, combine like terms. Group the x2-tiles, the x-tiles,
and the 1-tiles.
+
+ +
+
+
+ + + +
+
+
=
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MODELING ADDITION OF POLYNOMIALSYou can use algebra tiles to add the polynomials x2+ 4x+ 2and 2x23x1.
+ + + + + +
+
+ +
x2+ 4x+ 2 2x23x1
2 To add the polynomials, combine like terms. Group the x2-tiles, the x-tiles,
and the 1-tiles.
+
+ +
+
+
+ + + +
+
+
=
3 Find and remove the zero pairs.
The sum is 3x2+ x+ 1.
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An expression which is the sum of terms of the form axkwhere kis a nonnegative
integer is a polynomial. Polynomials are usually written in standard form.
Add ing and Subtract ing Polynom ials
Standard form means that the terms of the polynomial are placed in descending
order, from largest degree to smallest degree.
The degreeof each term of a polynomial is the exponent of the variable.
Polynomial in standard form:
2x3+ 5x24x+ 7
Degree Constant termLeading coefficient
The degree of a polynomialis the largest degree of its terms. When a
polynomial is written in standard form, the coefficient of the first term is
the leading coefficient.
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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:
Class i fy ing Polynom ials
Polynomial Degree
Classified by
degree
Classified by
number of terms
6
2x
3x+ 1
x2+ 2x5
4x38x
2x47x35x+ 1
0
1
1
4
2
3
constant
linear
linear
quartic
quadratic
cubic
monomial
monomial
binomial
polynomial
trinomial
binomial
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Find the sum. Write the answer in standard format.(5x3
x+ 2x2+ 7) + (3x2+ 7
4x) + (4x2
8
x3)
Adding Polynom ials
SOLUTION
Vertical format: Write each expression in standard form. Align like terms.
5x3+ 2x2 x+ 7
3x24x+ 7
x3 + 4x2 8+
4x3+ 9x25x+ 6
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Find the sum. Write the answer in standard format.(2x2+ x
5) + (x+ x2+ 6)
Adding Polynom ials
SOLUTION
Horizontal format: Add like terms.
(2x2+ x5) + (x+ x2+ 6) = (2x2+x2) + (x+ x) + (5+ 6)
= 3x2+ 2x+ 1
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Find the difference.(
2x3+ 5x2
x+ 8)
(
2x2+ 3x4)
Subtract ing Polynom ials
SOLUTION
Use a vertical format. To subtract, you add the opposite. This means you
multiply each term in the subtracted polynomial by
1and add.
2x3+ 5x2 x+ 8
2x3 + 3x4 Add the oppos i te
No change2x3+ 5x2 x+ 8
2x3 3x+ 4+
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Find the difference.(
2x3+ 5x2
x+ 8)
(
2x2+ 3x4)
Subtract ing Polynom ials
SOLUTION
Use a vertical format. To subtract, you add the opposite. This means you
multiply each term in the subtracted polynomial by
1and add.
2x3+ 5x2 x+ 8
2x3 + 3x4
5x2
4x+ 12
2x3+ 5x2 x+ 8
2x3 3x+ 4+
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Find the difference.(3x2
5x+ 3)
(2x2
x
4)
Subtract ing Polynom ials
SOLUTION
Use a horizontal format.
(3x25x+ 3)(2x2x4) = (3x25x+ 3) + (1)(2x2x4)
=x2
4x+ 7
= (3x25x+ 3)2x2+ x+ 4
= (3x22x2)+ (5x+x) + (3+4)
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Total Area= (10x)(14x2) (square inches)
Area of photo=
You are enlarging a 5-inch by 7-inch photo by a scale factor of xand mounting it on
a mat. You want the mat to be twice as wide as the enlarged photo and 2inches
less than twice as high as the enlarged photo.
Using Polyn om ials in Real Life
Write a model for the area of the mat around the photograph as a function of the
scale 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
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(10x)(14x2)(5x)(7x)
You are enlarging a 5-inch by 7-inch photo by a scale factor of xand mounting it on
a mat. You want the mat to be twice as wide as the enlarged photo and 2inches
less than twice as high as the enlarged photo.
Using Polyn om ials in Real Life
Write a model for the area of the mat around the photograph as a function of the
scale factor.
A =
= 140x220x35x2
SOLUTION
= 105x220x
A model for the area of the mat around the photograph as a function of the
scale factor xis A= 105x220x.
Algebraic
Model
5x
7x
14x
2
10x