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A table is shown. The top row is titled “Monomials” and lists the following monomials: 5, 4 b squared, negative 9 x cubed, negative 18. The next row is titled “Degree” and lists, in blue, 0, 2, 3, and 0. The next row is titled “Binomial” and lists the following binomials: b plus 1, 3a minus 7, y squared minus 9, 17 x cubed plus 14 x squared. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 1, 0, 1, 0, 2, 0, 3, 2 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 1, 1, 2, 3 in red. The next row is titled “Trinomial” and lists the following trinomials: x squared minus 5x plus 6, 4 y squared minus 7y plus 2, 5 a to the fourth minus 3 a cubed plus a, and x to the fourth plus 2 x squared minus 5. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 2, 1, 0, 2, 1, 0, 4, 3, 1, 4, 2, 0 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 2, 2, 4, 4 in red. The next row is titled “Polynomial” and lists the following polynomials: b plus 1, 4 y squared minus 7y plus 2, and 4 x to the fourth plus x cubed plus 8 x squared minus 9x plus 1. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 1, 0, 2, 1, 0, 4, 3, 2, 1, 0 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 1, 2, 4 in red.

Find the degree of the following polynomials:

  1. 4 x
  2. 3 x 3 5 x + 7
  3. −11
  4. −6 x 2 + 9 x 3
  5. 8 x + 2

Solution

4 x
The exponent of x is one. x = x 1 The degree is 1.
3 x 3 5 x + 7
The highest degree of all the terms is 3. The degree is 3
11
The degree of a constant is 0. The degree is 0.
−6 x 2 + 9 x 3
The highest degree of all the terms is 2. The degree is 2.
8 x + 2
The highest degree of all the terms is 1. The degree is 1.
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Find the degree of the following polynomials:

  1. −6 y
  2. 4 x 1
  3. 3 x 4 + 4 x 2 8
  4. 2 y 2 + 3 y + 9
  5. −18

  1. 1
  2. 1
  3. 4
  4. 2
  5. 0

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Find the degree of the following polynomials:

  1. 47
  2. 2 x 2 8 x + 2
  3. x 4 16
  4. y 5 5 y 3 + y
  5. 9 a 3

  1. 0
  2. 2
  3. 4
  4. 5
  5. 3

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Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form . Look back at the polynomials in [link] . Notice that they are all written in standard form. Get in the habit of writing the term with the highest degree first.

Add and subtract monomials

In The Language of Algebra , you simplified expressions by combining like terms. Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. Recall that when combining like terms only the coefficients are combined, never the exponents.

Add: 17 x 2 + 6 x 2 .

Solution

17 x 2 + 6 x 2
Combine like terms. 23 x 2
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Add: 12 x 2 + 5 x 2 .

17 x 2

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Add: −11 y 2 + 8 y 2 .

−3 y 2

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Subtract: 11 n ( −8 n ) .

Solution

11 n ( −8 n )
Combine like terms. 19 n
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Subtract: 9 n ( −5 n ) .

14 n

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Subtract: −7 a 3 ( −5 a 3 ) .

−2 a 3

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Simplify: a 2 + 4 b 2 7 a 2 .

Solution

a 2 + 4 b 2 7 a 2
Combine like terms. −6 a 2 + 4 b 2

Remember, −6 a 2 and 4 b 2 are not like terms. The variables are not the same.

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Add: 3 x 2 + 3 y 2 5 x 2 .

−2 x 2 + 3 y 2

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Add: 2 a 2 + b 2 4 a 2 .

−2 a 2 + b 2

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Add and subtract polynomials

Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms—those with the same variables with the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together. It may also be helpful to underline, circle, or box like terms.

Find the sum: ( 4 x 2 5 x + 1 ) + ( 3 x 2 8 x 9 ) .

Solution

.
Identify like terms. .
Rearrange to get the like terms together. .
Combine like terms. .
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Find the sum: ( 3 x 2 2 x + 8 ) + ( x 2 6 x + 2 ) .

4 x 2 − 8 x + 10

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Find the sum: ( 7 y 2 + 4 y 6 ) + ( 4 y 2 + 5 y + 1 ) .

11 y 2 + 9 y − 5

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Parentheses are grouping symbols. When we add polynomials as we did in [link] , we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.

Find the difference: ( 7 u 2 5 u + 3 ) ( 4 u 2 2 ) .

Solution

.
Distribute and identify like terms. .
Rearrange the terms. .
Combine like terms. .
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Find the difference: ( 6 y 2 + 3 y 1 ) ( 3 y 2 4 ) .

3 y 2 + 3 y + 3

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Find the difference: ( 8 u 2 7 u 2 ) ( 5 u 2 6 u 4 ) .

3 u 2 u + 2

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Subtract: ( m 2 3 m + 8 ) from ( 9 m 2 7 m + 4 ) .

Solution

.
Distribute and identify like terms. .
Rearrange the terms. .
Combine like terms. .
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Subtract: ( 4 n 2 7 n 3 ) from ( 8 n 2 + 5 n 3 ) .

4 n 2 + 12 n

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Subtract: ( a 2 4 a 9 ) from ( 6 a 2 + 4 a 1 ) .

5 a 2 + 8 a + 8

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Evaluate a polynomial for a given value

In The Language of Algebra we evaluated expressions. Since polynomials are expressions, we'll follow the same procedures to evaluate polynomials—substitute the given value for the variable into the polynomial, and then simplify.

Practice Key Terms 7

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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