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Mathematics LibreTexts

6.1: Add and Subtract Polynomials

  • Page ID
    18962
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    Learning Objectives

    By the end of this section, you will be able to:

    • Identify polynomials, monomials, binomials, and trinomials
    • Determine the degree of polynomials
    • Add and subtract monomials
    • Add and subtract polynomials
    • Evaluate a polynomial for a given value
    Quiz

    Before you get started, take this readiness quiz.

    1. Simplify: \(8x+3x\).
      If you missed this problem, review [link].
    2. Subtract: \((5n+8)−(2n−1)\).
      If you missed this problem, review [link].
    3. Write in expanded form: \(a^{5}\).
      If you missed this problem, review [link].

    Identify Polynomials, Monomials, Binomials and Trinomials

    You have learned that a term is a constant or the product of a constant and one or more variables. When it is of the form \(ax^{m}\), where \(a\) is a constant and \(m\) is a whole number, it is called a monomial. Some examples of monomial are \(8,−2x^{2},4y^{3}\), and \(11z^{7}\).

    Definition: Monomials

    A monomial is a term of the form \(ax^{m}\), where \(a\) is a constant and \(m\) is a positive whole number.

    A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

    Definitions: Polynomials
    • polynomial—A monomial, or two or more monomials combined by addition or subtraction, is a polynomial.
    • monomial—A polynomial with exactly one term is called a monomial.
    • binomial—A polynomial with exactly two terms is called a binomial.
    • trinomial—A polynomial with exactly three terms is called a trinomial.

    Here are some examples of polynomials.

    \[\begin{array}{lllll}{\text { Polynomial }} & {b+1} &{4 y^{2}-7 y+2} & {4 x^{4}+x^{3}+8 x^{2}-9 x+1} \\ {\text { Monomial }} & {14} & {8 y^{2}} & {-9 x^{3} y^{5}} & {-13}\\ {\text { Binomial }} & {a+7}&{4 b-5} & {y^{2}-16}& {3 x^{3}-9 x^{2}} \\ {\text { Trinomial }} & {x^{2}-7 x+12} & {9 y^{2}+2 y-8} & {6 m^{4}-m^{3}+8 m}&{z^{4}+3 z^{2}-1} \end{array} \nonumber\]

    Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

    Exercise \(\PageIndex{1}\)

    Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

    1. \(4y^{2}−8y−6\)
    2. \(−5a^{4}b^{2}\)
    3. \(2x^{5}−5x^{3}−3x + 4\)
    4. \(13−5m^{3}\)
    5. q
    Answer

    \(\begin{array}{lll}&{\text { Polynomial }} & {\text { Number of terms }} & {\text { Type }} \\ {\text { (a) }} & {4 y^{2}-8 y-6} & {3} & {\text { Trinomial }} \\ {\text { (b) }} & {-5 a^{4} b^{2}} & {1} & {\text { Monomial }} \\ {\text { (c) }} & {2 x^{5}-5 x^{3}-9 x^{2}+3 x+4} & {5} & {\text { Ponomial }} \\ {\text { (d) }} & {13-5 m^{3}} & {2} & {\text { Binomial }} \\ {\text { (e) }} & {q} & {1} & {\text { Monomial }}\end{array}\)

    Exercise \(\PageIndex{2}\)

    Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

    1. 5b
    2. \(8 y^{3}-7 y^{2}-y-3\)
    3. \(-3 x^{2}-5 x+9\)
    4. \(81-4 a^{2}\)
    5. \(-5 x^{6}\)
    Answer
    1. monomial
    2. polynomial
    3. trinomial
    4. binomial
    5. monomial
    Exercise \(\PageIndex{3}\)

    Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

    1. \(27 z^{3}-8\)
    2. \(12 m^{3}-5 m^{2}-2 m\)
    3. \(\frac{5}{6}\)
    4. \(8 x^{4}-7 x^{2}-6 x-5\)
    5. \(-n^{4}\)
    Answer
    1. binomial
    2. trinomial
    3. monomial
    4. polynomial
    5. monomial

    Determine the Degree of Polynomials

    The degree of a polynomial and the degree of its terms are determined by the exponents of the variable. A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0, i.e., it has no variable.

    Definition: Degree of a Polynomial
    • The degree of a term is the sum of the exponents of its variables.
    • The degree of a constant is 0.
    • The degree of a polynomial is the highest degree of all its terms.

    Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

    This table has 11 rows and 5 columns. The first column is a header column, and it names each row. The first row is named “Monomial,” and each cell in this row contains a different monomial. The second row is named “Degree,” and each cell in this row contains the degree of the monomial above it. The degree of 14 is 0, the degree of 8y squared is 2, the degree of negative 9x cubed y to the fifth power is 8, and the degree of negative 13a is 1. The third row is named “Binomial,” and each cell in this row contains a different binomial. The fourth row is named “Degree of each term,” and each cell contains the degrees of the two terms in the binomial above it. The fifth row is named “Degree of polynomial,” and each cell contains the degree of the binomial as a whole.” The degrees of the terms in a plus 7 are 0 and 1, and the degree of the whole binomial is 1. The degrees of the terms in 4b squared minus 5b are 2 and 1, and the degree of the whole binomial is 2. The degrees of the terms in x squared y squared minus 16 are 4 and 0, and the degree of the whole binomial is 4. The degrees of the terms in 3n cubed minus 9n squared are 3 and 2, and the degree of the whole binomial is 3. The sixth row is named “Trinomial,” and each cell in this row contains a different trinomial. The seventh row is named “Degree of each term,” and each cell contains the degrees of the three terms in the trinomial above it. The eighth row is named “Degree of polynomial,” and each cell contains the degree of the trinomial as a whole. The degrees of the terms in x squared minus 7x plus 12 are 2, 1, and 0, and the degree of the whole trinomial is 2. The degrees of the terms in 9a squared plus 6ab plus b squared are 2, 2, and 2, and the degree of the trinomial as a whole is 2. The degrees of the terms in 6m to the fourth power minus m cubed n squared plus 8mn to the fifth power are 4, 5, and 6, and the degree of the whole trinomial is 6. The degrees of the terms in z to the fourth power plus 3z squared minus 1 are 4, 2, and 0, and the degree of the whole trinomial is 4. The ninth row is named “Polynomial,” and each cell contains a different polynomial. The tenth row is named “Degree of each term,” and the eleventh row is named “Degree of polynomial.” The degrees of the terms in b plus 1 are 1 and 0, and the degree of the whole polynomial is 1. The degrees of the terms in 4y squared minus 7y plus 2 are 2, 1, and 0, and the degree of the whole polynomial is 2. The degrees of the terms in 4x to the fourth power plus x cubed plus 8x squared minus 9x plus 1 are 4, 3, 2, 1, and 0, and the degree of the whole polynomial is 4.

    A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

    Exercise \(\PageIndex{4}\)

    Find the degree of the following polynomials.

    1. 10y
    2. \(4 x^{3}-7 x+5\)
    3. −15
    4. \(-8 b^{2}+9 b-2\)
    5. \(8 x y^{2}+2 y\)
    Answer
    1. \(\begin{array}{ll} & 10y\\ \text{The exponent of y is one. } y=y^1 & \text{The degree is 1.}\end{array}\)
    2. \(\begin{array}{ll} & 4 x^{3}-7 x+5\\ \text{The highest degree of all the terms is 3.} & \text{The degree is 3.}\end{array}\)
    3. \(\begin{array}{ll} & -15\\ \text{The degree of a constant is 0.} & \text{The degree is 0.}\end{array}\)
    4. \(\begin{array}{ll} & -8 b^{2}+9 b-2\\ \text{The highest degree of all the terms is 2.} & \text{The degree is 2.}\end{array}\)
    5. \(\begin{array}{ll} & 8 x y^{2}+2 y\\ \text{The highest degree of all the terms is 3.} & \text{The degree is 3.}\end{array}\)
    Exercise \(\PageIndex{5}\)

    Find the degree of the following polynomials:

    1. −15b
    2. \(10 z^{4}+4 z^{2}-5\)
    3. \(12 c^{5} d^{4}+9 c^{3} d^{9}-7\)
    4. \(3 x^{2} y-4 x\)
    5. −9
    Answer
    1. 1
    2. 4
    3. 12
    4. 3
    5. 0
    Exercise \(\PageIndex{6}\)

    Find the degree of the following polynomials:

    1. 52
    2. \(a^{4} b-17 a^{4}\)
    3. \(5 x+6 y+2 z\)
    4. \(3 x^{2}-5 x+7\)
    5. \(-a^{3}\)
    Answer
    1. 0
    2. 5
    3. 1
    4. 2
    5. 3

    Add and Subtract Monomials

    You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

    Exercise \(\PageIndex{7}\)

    Add:\(25 y^{2}+15 y^{2}\)

    Answer

    \(\begin{array}{ll} & 25 y^{2}+15 y^{2}\\ \text{Combine like terms.} & 40y^{2}\end{array}\)

    Exercise \(\PageIndex{8}\)

    Add: \(12 q^{2}+9 q^{2}\)

    Answer

    21\(q^{2}\)

    Exercise \(\PageIndex{9}\)

    Add:\(-15 c^{2}+8 c^{2}\)

    Answer

    \(-7 c^{2}\)

    Exercise \(\PageIndex{10}\)

    Subtract: 16p−(−7p)

    Answer

    \(\begin{array}{ll} & 16p−(−7p) \\ \text{Combine like terms.} & 23p\end{array}\)

    Exercise \(\PageIndex{11}\)

    Subtract: 8m−(−5m).

    Answer

    13m

    Exercise \(\PageIndex{12}\)

    Subtract: \(-15 z^{3}-\left(-5 z^{3}\right)\)

    Answer

    \(-10 z^{3}\)

    Remember that like terms must have the same variables with the same exponents.

    Exercise \(\PageIndex{13}\)

    Simplify: \(c^{2}+7 d^{2}-6 c^{2}\)

    Answer

    \(\begin{array}{ll} & c^{2}+7 d^{2}-6 c^{2} \\ \text{Combine like terms.} & -5 c^{2}+7 d^{2} \end{array}\)

    Exercise \(\PageIndex{14}\)

    Add: \(8 y^{2}+3 z^{2}-3 y^{2}\)

    Answer

    \(5 y^{2}+3 z^{2}\)

    Exercise \(\PageIndex{15}\)

    Add: \(3 m^{2}+n^{2}-7 m^{2}\)

    Answer

    \(-4 m^{2}+n^{2}\)

    Exercise \(\PageIndex{16}\)

    Simplify: \(u^{2} v+5 u^{2}-3 v^{2}\)

    Answer

    \(\begin{array}{ll} &u^{2} v+5 u^{2}-3 v^{2}
    \\ \text{There are no like terms to combine.} & u^{2} v+5 u^{2}-3 v^{2} \end{array}\)

    Exercise \(\PageIndex{17}\)

    Simplify: \(m^{2} n^{2}-8 m^{2}+4 n^{2}\)

    Answer

    There are no like terms to combine.

    Exercise \(\PageIndex{18}\)

    Simplify: \(p q^{2}-6 p-5 q^{2}\)

    Answer

    There are no like terms to combine.

    Add and Subtract Polynomials

    We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

    Exercise \(\PageIndex{19}\)

    Find the sum: \(\left(5 y^{2}-3 y+15\right)+\left(3 y^{2}-4 y-11\right)\)

    Answer
    Identify like terms. 5 y squared minus 3 y plus 15, plus 3 y squared minus 4 y minus 11.
    Rearrange to get the like terms together. 5y squared plus 3y squared, identified as like terms, minus 3y minus 4y, identified as like terms, plus 15 minus 11, identified as like terms.
    Combine like terms. 8 y squared minus 7y plus 4.
    Exercise \(\PageIndex{20}\)

    Find the sum: \(\left(7 x^{2}-4 x+5\right)+\left(x^{2}-7 x+3\right)\)

    Answer

    \(8 x^{2}-11 x+1\)

    Exercise \(\PageIndex{21}\)

    Find the sum:\(\left(14 y^{2}+6 y-4\right)+\left(3 y^{2}+8 y+5\right)\)

    Answer

    \(17 y^{2}+14 y+1\)

    Exercise \(\PageIndex{22}\)

    Find the difference: \(\left(9 w^{2}-7 w+5\right)-\left(2 w^{2}-4\right)\)

    Answer
      9 w squared minus 7 w plus 5, minus 2 w squared minus 4.
    Distribute and identify like terms. 9 w squared and 2 w squared are like terms. 5 and 4 are also like terms.
    Rearrange the terms. 9 w squared minus 2 w squared minus 7 w plus 5 plus 4.
    Combine like terms. 7 w squared minus 7 w plus 9.
    Exercise \(\PageIndex{23}\)

    Find the difference: \(\left(8 x^{2}+3 x-19\right)-\left(7 x^{2}-14\right)\)

    Answer

    \(15 x^{2}+3 x-5\)

    Exercise \(\PageIndex{24}\)

    Find the difference: \(\left(9 b^{2}-5 b-4\right)-\left(3 b^{2}-5 b-7\right)\)

    Answer

    \(6 b^{2}+3\)

    Exercise \(\PageIndex{25}\)

    Subtract: \(\left(c^{2}-4 c+7\right)\) from \(\left(7 c^{2}-5 c+3\right)\)

    Answer
      .
      7 c squared minus 5 c plus 3, minus c squared minus 4c plus 7.
    Distribute and identify like terms. 7 c squared and c squared are like terms. Minus 5c and 4c are like terms. 3 and minus 7 are like terms.
    Rearrange the terms. 7 c squared minus c squared minus 5 c plus 4 c plus 3 minus 7.
    Combine like terms. 6 c squared minus c minus 4.
    Exercise \(\PageIndex{26}\)

    Subtract: \(\left(5 z^{2}-6 z-2\right)\) from \(\left(7 z^{2}+6 z-4\right)\)

    Answer

    \(2 z^{2}+12 z-2\)

    Exercise \(\PageIndex{27}\)

    Subtract: \(\left(x^{2}-5 x-8\right)\) from \(\left(6 x^{2}+9 x-1\right)\)

    Answer

    \(5 x^{2}+14 x+7\)

    Exercise \(\PageIndex{28}\)

    Find the sum: \(\left(u^{2}-6 u v+5 v^{2}\right)+\left(3 u^{2}+2 u v\right)\)

    Answer

    \(\begin{array} {ll} & {\left(u^{2}-6 u v+5 v^{2}\right)+\left(3 u^{2}+2 u v\right)} \\\text{Distribute.} & {u^{2}-6 u v+5 v^{2}+3 u^{2}+2 u v} \\ \text{Rearrange the terms, to put like terms together} & {u^{2}+3 u^{2}-6 u v+2 u v+5 v^{2}} \\ \text{Combine like terms.} & {4 u^{2}-4 u v+5 v^{2}}\end{array}\)

    Exercise \(\PageIndex{29}\)

    Find the sum: \(\left(3 x^{2}-4 x y+5 y^{2}\right)+\left(2 x^{2}-x y\right)\)

    Answer

    \(5 x^{2}-5 x y+5 y^{2}\)

    Exercise \(\PageIndex{30}\)

    Find the sum: \(\left(2 x^{2}-3 x y-2 y^{2}\right)+\left(5 x^{2}-3 x y\right)\)

    Answer

    \(7 x^{2}-6 x y-2 y^{2}\)

    Exercise \(\PageIndex{31}\)

    Find the difference: \(\left(p^{2}+q^{2}\right)-\left(p^{2}+10 p q-2 q^{2}\right)\)

    Answer

    \(\begin{array}{ll} & {\left(p^{2}+q^{2}\right)-\left(p^{2}+10 p q-2 q^{2}\right)} \\ \text{Distribute.} &{p^{2}+q^{2}-p^{2}-10 p q+2 q^{2}} \\\text{Rearrange the terms, to put like terms together} & {p^{2}-p^{2}-10 p q+q^{2}+2 q^{2}} \\\text{Combine like terms.} & {-10 p q^{2}+3 q^{2}}\end{array}\)

    Exercise \(\PageIndex{32}\)

    Find the difference: \(\left(a^{2}+b^{2}\right)-\left(a^{2}+5 a b-6 b^{2}\right)\)

    Answer

    \(-5 a b-5 b^{2}\)

    Exercise \(\PageIndex{33}\)

    Find the difference: \(\left(m^{2}+n^{2}\right)-\left(m^{2}-7 m n-3 n^{2}\right)\)

    Answer

    \(4 n^{2}+7 m n\)

    Exercise \(\PageIndex{34}\)

    Simplify: \(\left(a^{3}-a^{2} b\right)-\left(a b^{2}+b^{3}\right)+\left(a^{2} b+a b^{2}\right)\)

    Answer

    \(\begin{array}{ll } & {\left(a^{3}-a^{2} b\right)-\left(a b^{2}+b^{3}\right)+\left(a^{2} b+a b^{2}\right)} \\ \text{Distribute.} &{a^{3}-a^{2} b-a b^{2}-b^{3}+a^{2} b+a b^{2}} \\ \text{Rearrange the terms, to put like terms together} & {a^{3}-a^{2} b+a^{2} b-a b^{2}+a b^{2}-b^{3}} \\ \text{Combine like terms.} &{a^{3}-b^{3}}\end{array}\)

    Exercise \(\PageIndex{35}\)

    Simplify: \(\left(x^{3}-x^{2} y\right)-\left(x y^{2}+y^{3}\right)+\left(x^{2} y+x y^{2}\right)\)

    Answer

    \(x^{3}-y^{3}\)

    Exercise \(\PageIndex{36}\)

    Simplify: \(\left(p^{3}-p^{2} q\right)+\left(p q^{2}+q^{3}\right)-\left(p^{2} q+p q^{2}\right)\)

    Answer

    \(p^{3}-2 p^{2} q+q^{3}\)

    Evaluate a Polynomial for a Given Value

    We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.

    Exercise \(\PageIndex{37}\)

    Evaluate \(5x^{2}−8x+4\) when

    1. x=4
    2. x=−2
    3. x=0
    Answer
    1. x=4  
      5 x squared minus 8 x plus 4.
    Substitute 4 for x. 5 times 4 squared minus 8 times 4 plus 4.
    Simplify the exponents. 5 times 16 minus 8 times 4 plus 4.
    Multiply. 80 minus 32 plus 4.
    Simplify. 52.
    2. x=−2  
      5 x squared minus 8 x plus 4.
    Substitute negative 2 for x. 5 times negative 2 squared minus 8 times negative 2 plus 4.
    Simplify the exponents. 5 times 4 minus 8 times negative 2 plus 4.
    Multiply. 20 plus 16 plus 4.
    Simplify. 40.
    3. x=0  
      5 x squared minus 8 x plus 4.
    Substitute 0 for x. 5 times 0 squared minus 8 times 0 plus 4.
    Simplify the exponents. 5 times 0 minus 8 times 0 plus 4.
    Multiply. 0 plus 0 plus 4.
    Simplify. 4.
     
    Exercise \(\PageIndex{38}\)

    Evaluate: \(3x^{2}+2x−15\) when

    1. x=3
    2. x=−5
    3. x=0
    Answer
    1. 18
    2. 50
    3. −15
    Exercise \(\PageIndex{39}\)

    Evaluate: \(5z^{2}−z−4\) when

    1. z=−2
    2. z=0
    3. z=2
    Answer
    1. 18
    2. −4
    3. 14
    Exercise \(\PageIndex{40}\)

    The polynomial \(−16t^{2}+250\) gives the height of a ball tt seconds after it is dropped from a 250 foot tall building. Find the height after t=2 seconds.

    Answer

    \(\begin{array}{ll } & −16t^{2}+250 \\ \text{Substitute t = 2.} & -16(2)^{2} + 250 \\ \text{Simplify }& −16\cdot 4+250 \\ \text{Simplify }& -64 + 250\\ \text{Simplify }& 186 \\& \text{After 2 seconds the height of the ball is 186 feet. } \end{array}\)

    Exercise \(\PageIndex{41}\)

    The polynomial \(−16t^{2}+250\) gives the height of a ball tt seconds after it is dropped from a 250 foot tall building. Find the height after t=0 seconds.

    Answer

    250

    Exercise \(\PageIndex{42}\)

    The polynomial \(−16t^{2}+250\) gives the height of a ball tt seconds after it is dropped from a 250 foot tall building. Find the height after t=3 seconds.

    Answer

    106

    Exercise \(\PageIndex{43}\)

    The polynomial \(6x^{2}+15xy\) gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with x=4 feet and y=6y=6 feet.

    Answer
      6 x squared plus 15 x y.
    Substitute x equals 4 and y equals 6. 6 times 4 squared plus 15 times 4 times 6.
    Simplify. 6 times 16 plus 15 times 4 times 6.
    Simplify. 96 plus 360.
    Simplify. 456.
      The cost of producing the box is $456.
    Exercise \(\PageIndex{43}\)

    The polynomial \(6x^{2}+15xy\) gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with x=6 feet and y=4 feet.

    Answer

    $576

    Exercise \(\PageIndex{44}\)

    The polynomial \(6x^{2}+15xy\) gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with x=5 feet and y=8 feet.

    Answer

    $750

    Key Concepts

    • Monomials
      • A monomial is a term of the form \(ax^{m}\), where aa is a constant and mm is a whole number
         
    • Polynomials
      • polynomial—A monomial, or two or more monomials combined by addition or subtraction is a polynomial.
      • monomial—A polynomial with exactly one term is called a monomial.
      • binomial—A polynomial with exactly two terms is called a binomial.
      • trinomial—A polynomial with exactly three terms is called a trinomial.
         
    • Degree of a Polynomial
      • The degree of a term is the sum of the exponents of its variables.
      • The degree of a constant is 0.
      • The degree of a polynomial is the highest degree of all its terms.

    Glossary

    binomial
    A binomial is a polynomial with exactly two terms.
    degree of a constant
    The degree of any constant is 0.
    degree of a polynomial
    The degree of a polynomial is the highest degree of all its terms.
    degree of a term
    The degree of a term is the exponent of its variable.
    monomial
    A monomial is a term of the form \(ax^m\), where a is a constant and m is a whole number; a monomial has exactly one term.
    polynomial
    A polynomial is a monomial, or two or more monomials combined by addition or subtraction.
    standard form
    A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.
    trinomial
    A trinomial is a polynomial with exactly three terms.