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1.6 Polynomials and Their Operations

Learning Objectives

  1. Identify a polynomial and determine its degree.
  2. Add and subtract polynomials.
  3. Multiply and divide polynomials.

Definitions

A polynomialAn algebraic expression consisting of terms with real number coefficients and variables with whole number exponents. is a special algebraic expression with terms that consist of real number coefficients and variable factors with whole number exponents. Some examples of polynomials follow:

3x2

7xy+5

32x3+3x212x+1

6x2y4xy3+7

The degree of a termThe exponent of the variable. If there is more than one variable in the term, the degree of the term is the sum their exponents. in a polynomial is defined to be the exponent of the variable, or if there is more than one variable in the term, the degree is the sum of their exponents. Recall that x0=1; any constant term can be written as a product of x0 and itself. Hence the degree of a constant term is 0.

Term

Degree

3x2

2

6x2y

2+1=3

7a2b3

2+3=5

8

0, since 8=8x0

2x

1, since 2x=2x1

The degree of a polynomialThe largest degree of all of its terms. is the largest degree of all of its terms.

Polynomial

Degree

4x53x3+2x1

5

6x2y5xy3+7

4, because 5xy3 has degree 4.

12x+54

1, because 12x=12x1

Of particular interest are polynomials with one variableA polynomial where each term has the form anxn, where an is any real number and n is any whole number., where each term is of the form anxn. Here an is any real number and n is any whole number. Such polynomials have the standard form:

anxn+an1xn1++a1x+a0

Typically, we arrange terms of polynomials in descending order based on the degree of each term. The leading coefficientThe coefficient of the term with the largest degree. is the coefficient of the variable with the highest power, in this case, an.

Example 1

Write in standard form: 3x4x2+5x3+72x4.

Solution:

Since terms are defined to be separated by addition, we write the following:

3x4x2+5x3+72x4=3x+(4)x2+5x3+7+(2)x4

In this form, we can see that the subtraction in the original corresponds to negative coefficients. Because addition is commutative, we can write the terms in descending order based on the degree as follows:

=(2)x4+5x3+(4)x2+3x+7=2x4+5x34x2+3x+7

Answer: 2x4+5x34x2+3x+7

We classify polynomials by the number of terms and the degree:

Expression

Classification

Degree

5x7

Monomial (one term)

7

8x61

Binomial (two terms)

6

3x2+x1

Trinomial (three terms)

2

5x32x2+3x6

Polynomial (many terms)

3

Polynomial with one term.

Polynomial with two terms.

Polynomial with three terms.

We can further classify polynomials with one variable by their degree:

Polynomial

Name

5

Constant (degree 0)

2x+1

Linear (degree 1)

3x2+5x3

Quadratic (degree 2)

x3+x2+x+1

Cubic (degree 3)

7x4+3x37x+8

Fourth-degree polynomial

A polynomial with degree 0.

A polynomial with degree 1.

A polynomial with degree 2.

A polynomial with degree 3.

In this text, we call any polynomial of degree n4 an nth-degree polynomial. In other words, if the degree is 4, we call the polynomial a fourth-degree polynomial. If the degree is 5, we call it a fifth-degree polynomial, and so on.

Example 2

State whether the following polynomial is linear or quadratic and give the leading coefficient: 25+4xx2.

Solution:

The highest power is 2; therefore, it is a quadratic polynomial. Rewriting in standard form we have

x2+4x+25

Here x2=1x2 and thus the leading coefficient is −1.

Answer: Quadratic; leading coefficient: −1

Adding and Subtracting Polynomials

We begin by simplifying algebraic expressions that look like +(a+b) or (a+b). Here, the coefficients are actually implied to be +1 and −1 respectively and therefore the distributive property applies. Multiply each term within the parentheses by these factors as follows:

+(a+b)=+1(a+b)=(+1)a+(+1)b=a+b(a+b)=1(a+b)=(1)a+(1)b=ab

Use this idea as a means to eliminate parentheses when adding and subtracting polynomials.

Example 3

Add: 9x2+(x25).

Solution:

The property +(a+b)=a+b allows us to eliminate the parentheses, after which we can then combine like terms.

9x2+(x25)=9x2+x25=10x25

Answer: 10x25

Example 4

Add: (3x2y24xy+9)+(2x2y26xy7).

Solution:

Remember that the variable parts have to be exactly the same before we can add the coefficients.

(3x2y24xy+9)+(2x2y26xy7)=3x2y24xy+9+2x2y26xy7=5x2y210xy+2

Answer: 5x2y210xy+2

When subtracting polynomials, the parentheses become very important.

Example 5

Subtract: 4x2(3x2+5x).

Solution:

The property (a+b)=ab allows us to remove the parentheses after subtracting each term.

4x2(3x2+5x)=4x23x25x=x25x

Answer: x25x

Subtracting a quantity is equivalent to multiplying it by −1.

Example 6

Subtract: (3x22xy+y2)(2x2xy+3y2).

Solution:

Distribute the −1, remove the parentheses, and then combine like terms. Multiplying the terms of a polynomial by −1 changes all the signs.

=3x22xy+y22x2+xy3y2=x2xy2y2

Answer: x2xy2y2

Try this! Subtract: (7a22ab+b2)(a22ab+5b2).

Answer: 6a24b2

Multiplying Polynomials

Use the product rule for exponents, xmxn=xm+n, to multiply a monomial times a polynomial. In other words, when multiplying two expressions with the same base, add the exponents. To find the product of monomials, multiply the coefficients and add the exponents of variable factors with the same base. For example,

7x48x3=78x4x3Commutativeproperty=56x4+3    Productruleforexponents=56x7

To multiply a polynomial by a monomial, apply the distributive property, and then simplify each term.

Example 7

Multiply: 5xy2(2x2y2xy+1).

Solution:

Apply the distributive property and then simplify.

=5xy22x2y25xy2xy+5xy21=10x3y45x2y3+5xy2

Answer: 10x3y45x2y3+5xy2

To summarize, multiplying a polynomial by a monomial involves the distributive property and the product rule for exponents. Multiply all of the terms of the polynomial by the monomial. For each term, multiply the coefficients and add exponents of variables where the bases are the same.

In the same manner that we used the distributive property to distribute a monomial, we use it to distribute a binomial. (a+b)(c+d)=(a+b)c+(a+b)d=ac+bc+ad+bd=ac+ad+bc+bd Here we apply the distributive property multiple times to produce the final result. This same result is obtained in one step if we apply the distributive property to a and b separately as follows:

This is often called the FOIL method. Multiply the first, outer, inner, and then last terms.

Example 8

Multiply: (6x1)(3x5).

Solution:

Distribute 6x and −1 and then combine like terms.

(6x1)(3x5)=6x3x6x5+(1)3x(1)5=18x230x3x+5=18x233x+5

Answer: 18x233x+5

Consider the following two calculations:

(a+b)2=(a+b)(a+b)=a2+ab+ba+b2=a2+ab+ab+b2=a2+2ab+b2

(ab)2=(ab)(ab)=a2abba+b2=a2abab+b2=a22ab+b2

This leads us to two formulas that describe perfect square trinomialsThe trinomials obtained by squaring the binomials (a+b)2=a2+2ab+b2 and (ab)2=a22ab+b2.:

(a+b)2=a2+2ab+b2(ab)2=a22ab+b2

We can use these formulas to quickly square a binomial.

Example 9

Multiply: (3x+5)2.

Solution:

Here a=3x and b=5. Apply the formula:

Answer: 9x2+30x+25

This process should become routine enough to be performed mentally. Our third special product follows: (a+b)(ab)=a2ab+bab2=a2ab+abb2=a2b2 This product is called difference of squaresThe special product obtained by multiplying conjugate binomials (a+b)(ab)=a2b2.:

(a+b)(ab)=a2b2

The binomials (a+b) and (ab) are called conjugate binomialsThe binomials (a+b) and (ab).. When multiplying conjugate binomials the middle terms are opposites and their sum is zero; the product is itself a binomial.

Example 10

Multiply: (3xy+1)(3xy1).

Solution:

(3xy+1)(3xy1)=(3xy)23xy+3xy12=9x2y21

Answer: 9x2y21

Try this! Multiply: (x2+5y2)(x25y2).

Answer: (x425y4)

Example 11

Multiply: (5x2)3.

Solution:

Here we perform one product at a time.

Answer: 125x2150x2+60x8

Dividing Polynomials

Use the quotient rule for exponents, xmxn=xmn, to divide a polynomial by a monomial. In other words, when dividing two expressions with the same base, subtract the exponents. In this section, we will assume that all variables in the denominator are nonzero.

Example 12

Divide: 24x7y58x3y2.

Solution:

Divide the coefficients and apply the quotient rule by subtracting the exponents of the like bases.

24x7y58x3y2=248x73y52=3x4y3

Answer: 3x4y3

When dividing a polynomial by a monomial, we may treat the monomial as a common denominator and break up the fraction using the following property: a+bc=ac+bc Applying this property will result in terms that can be treated as quotients of monomials.

Example 13

Divide: 5x4+25x315x25x2.

Solution:

Break up the fraction by dividing each term in the numerator by the monomial in the denominator, and then simplify each term.

5x4+25x315x25x2=5x45x2+25x35x215x25x2=55x42+255x32155x22=1x2+5x13x0=x2+5x31

Answer: x2+5x3

We can check our division by multiplying our answer, the quotient, by the monomial in the denominator, the divisor, to see if we obtain the original numerator, the dividend.

DividendDivisor=Quotient

5x4+25x315x25x2=x2+5x3

or

or

Dividend=DivisorQuotient

5x4+25x315x2=5x2(x2+5x3)

The same technique outlined for dividing by a monomial does not work for polynomials with two or more terms in the denominator. In this section, we will outline a process called polynomial long divisionThe process of dividing two polynomials using the division algorithm., which is based on the division algorithm for real numbers. For the sake of clarity, we will assume that all expressions in the denominator are nonzero.

Example 14

Divide: x3+3x28x4x2.

Solution:

Here x2 is the divisor and x3+3x28x4 is the dividend. To determine the first term of the quotient, divide the leading term of the dividend by the leading term of the divisor.

Multiply the first term of the quotient by the divisor, remembering to distribute, and line up like terms with the dividend.

Subtract the resulting quantity from the dividend. Take care to subtract both terms.

Bring down the remaining terms and repeat the process.

Notice that the leading term is eliminated and that the result has a degree that is one less. The complete process is illustrated below:

Polynomial long division ends when the degree of the remainder is less than the degree of the divisor. Here, the remainder is 0. Therefore, the binomial divides the polynomial evenly and the answer is the quotient shown above the division bar.

x3+3x28x4x2=x2+5x+2

To check the answer, multiply the divisor by the quotient to see if you obtain the dividend as illustrated below:

x3+3x28x4=(x2)(x2+5x+2)

This is left to the reader as an exercise.

Answer: x2+5x+2

Next, we demonstrate the case where there is a nonzero remainder.

Just as with real numbers, the final answer adds to the quotient the fraction where the remainder is the numerator and the divisor is the denominator. In general, when dividing we have: DividendDivisor=Quotient+RemainderDivisor If we multiply both sides by the divisor we obtain, Dividend=Quotient×Divisor+Remainder

Example 15

Divide: 6x25x+32x1.

Solution:

Since the denominator is a binomial, begin by setting up polynomial long division.

To start, determine what monomial times 2x1 results in a leading term 6x2. This is the quotient of the given leading terms: (6x2)÷(2x)=3x. Multiply 3x times the divisor 2x1, and line up the result with like terms of the dividend.

Subtract the result from the dividend and bring down the constant term +3.

Subtracting eliminates the leading term. Multiply 2x1 by −1 and line up the result.

Subtract again and notice that we are left with a remainder.

The constant term 2 has degree 0 and thus the division ends. Therefore,

6x25x+32x1=3x1+22x1

To check that this result is correct, we multiply as follows:

quotient×divisor+remainder=(3x1)(2x1)+2=6x23x2x+1+2=6x25x+2=dividend

Answer: 3x1+22x1

Occasionally, some of the powers of the variables appear to be missing within a polynomial. This can lead to errors when lining up like terms. Therefore, when first learning how to divide polynomials using long division, fill in the missing terms with zero coefficients, called placeholdersTerms with zero coefficients used to fill in all missing exponents within a polynomial..

Example 16

Divide: 27x3+643x+4.

Solution:

Notice that the binomial in the numerator does not have terms with degree 2 or 1. The division is simplified if we rewrite the expression with placeholders:

27x3+64=27x3+0x2+0x+64

Set up polynomial long division:

We begin with 27x3÷3x=9x2 and work the rest of the division algorithm.

Answer: 9x212x+16

Example 17

Divide: 3x42x3+6x2+23x7x22x+5.

Solution:

Begin the process by dividing the leading terms to determine the leading term of the quotient 3x4÷x2=3x2. Take care to distribute and line up the like terms. Continue the process until the remainder has a degree less than 2.

The remainder is x2. Write the answer with the remainder:

3x42x3+6x2+23x7x22x+5=3x2+4x1+x2x22x+5

Answer: 3x2+4x1+x2x22x+5

Polynomial long division takes time and practice to master. Work lots of problems and remember that you may check your answers by multiplying the quotient by the divisor (and adding the remainder if present) to obtain the dividend.

Try this! Divide: 6x413x3+9x214x+63x2.

Answer: 2x33x2+x423x2

Key Takeaways

  • Polynomials are special algebraic expressions where the terms are the products of real numbers and variables with whole number exponents.
  • The degree of a polynomial with one variable is the largest exponent of the variable found in any term. In addition, the terms of a polynomial are typically arranged in descending order based on the degree of each term.
  • When adding polynomials, remove the associated parentheses and then combine like terms. When subtracting polynomials, distribute the −1, remove the parentheses, and then combine like terms.
  • To multiply polynomials apply the distributive property; multiply each term in the first polynomial with each term in the second polynomial. Then combine like terms.
  • When dividing by a monomial, divide all terms in the numerator by the monomial and then simplify each term.
  • When dividing a polynomial by another polynomial, apply the division algorithm.

Topic Exercises

    Part A: Definitions

      Write the given polynomials in standard form.

    1. 1xx2

    2. y5+y2

    3. y3y2+5y3

    4. 812a2+a3a

    5. 2x2+6x5x3+x4

    6. a35+a2+2a4a5+6a

      Classify the given polynomial as a monomial, binomial, or trinomial and state the degree.

    1. x2x+2

    2. 510x3

    3. x2y2+5xy6

    4. 2x3y2

    5. x41

    6. 5

      State whether the polynomial is linear or quadratic and give the leading coefficient.

    1. 19x2

    2. 10x2

    3. 2x3

    4. 100x

    5. 5x2+3x1

    6. x1

    7. x62x2

    8. 15x

    Part B: Adding and Subtracting Polynomials

      Simplify.

    1. (5x23x2)+(2x26x+7)

    2. (x2+7x12)+(2x2x+3)

    3. (x2+5x+10)+(x210)

    4. (x21)+(4x+2)

    5. (10x2+3x2)(x26x+1)

    6. (x23x8)(2x23x8)

    7. (23x2+34x1)(16x2+52x12)

    8. (45x258x+106)(310x223x+35)

    9. (x2y2+7xy5)(2x2y2+5xy4)

    10. (x2y2)(x2+6xy+y2)

    11. (a2b2+5ab2)+(7ab2)(4a2b2)

    12. (a2+9ab6b2)(a2b2)+7ab

    13. (10x2y8xy+5xy2)(x2y4xy)+(xy2+4xy)

    14. (2m2n6mn+9mn2)(m2n+10mn)m2n

    15. (8x2y25xy+2)(x2y2+5)+(2xy3)

    16. (x2y2)(5x22xyy2)(x27xy)

    17. (16a22ab+34b2)(53a2+45b2)+118ab

    18. (52x22y2)(75x212xy+73y2)12xy

    19. (x2n+5xn2)+(2x2n3xn1)

    20. (7x2nxn+5)(6x2nxn8)

    21. Subtract 4y3 from y2+7y10.

    22. Subtract x2+3x2 from 2x2+4x1.

    23. A right circular cylinder has a height that is equal to the radius of the base, h=r. Find a formula for the surface area in terms of h.

    24. A rectangular solid has a width that is twice the height and a length that is 3 times that of the height. Find a formula for the surface area in terms of the height.

    Part C: Multiplying Polynomials

      Multiply.

    1. 8x22x

    2. 10x2y5x3y2

    3. 2x(5x1)

    4. 4x(3x5)

    5. 7x2(2x6)

    6. 3x2(x2x+3)

    7. 5y4(y22y+3)

    8. 52a3(24a26a+4)

    9. 2xy(x27xy+y2)

    10. 2a2b(a23ab+5b2)
    11. xn(x2+x+1)

    12. xn(x2nxn1)

    13. (x+4)(x5)

    14. (x7)(x6)

    15. (2x3)(3x1)

    16. (9x+1)(3x+2)

    17. (3x2y2)(x25y2)

    18. (5y2x2)(2y23x2)

    19. (3x+5)(3x5)

    20. (x+6)(x6)

    21. (a2b2)(a2+b2)

    22. (ab+7)(ab7)

    23. (4x5y2)(3x2y)

    24. (xy+5)(xy)

    25. (x5)(x23x+8)

    26. (2x7)(3x2x+1)

    27. (x2+7x1)(2x23x1)

    28. (4x2x+6)(5x24x3)

    29. (x+8)2

    30. (x3)2

    31. (2x5)2

    32. (3x+1)2

    33. (a3b)2

    34. (7ab)2

    35. (x2+2y2)2
    36. (x26y)2
    37. (a2a+5)2
    38. (x23x1)2
    39. (x3)3

    40. (x+2)3

    41. (3x+1)3

    42. (2x3)3

    43. (x+2)4

    44. (x3)4

    45. (2x1)4

    46. (3x1)4

    47. (x2n+5)(x2n5)
    48. (xn1)(x2n+4xn3)
    49. (x2n1)2
    50. (x3n+1)2
    51. Find the product of 3x2 and x25x2.

    52. Find the product of x2+4 and x31.

    53. Each side of a square measures 3x3 units. Determine the area in terms of x.

    54. Each edge of a cube measures 2x2 units. Determine the volume in terms of x.

    Part D: Dividing Polynomials

      Divide.

    1. 125x5y225x4y2
    2. 256x2y3z564x2yz2
    3. 20x312x2+4x4x
    4. 15x475x3+18x23x2
    5. 12a2b+28ab24ab4ab
    6. 2a4b3+16a2b2+8ab32ab2
    7. x3+x23x+9x+3
    8. x34x29x+20x5
    9. 6x311x2+7x62x3
    10. 9x39x2x+13x1
    11. 16x3+8x239x+174x3
    12. 12x356x2+55x+302x5
    13. 6x4+13x39x2x+63x+2
    14. 25x410x3+11x27x+15x1
    15. 20x4+12x3+9x2+10x+52x+1
    16. 25x445x326x2+36x115x2
    17. 3x4+x21x2
    18. x4+x3x+3
    19. x310x2
    20. x3+15x+3
    21. y5+1y+1
    22. y6+1y+1
    23. x44x3+6x27x1x2x+2
    24. 6x4+x32x2+2x+43x2x+1
    25. 2x37x2+8x3x22x+1
    26. 2x4+3x36x24x+3x2+x3
    27. x4+4x32x24x+1x21
    28. x4+x1x2+1
    29. x3+6x2y+4xy2y3x+y
    30. 2x33x2y+4xy23y3xy
    31. 8a3b32ab
    32. a3+27b3a+3b
    33. Find the quotient of 10x211x+3 and 2x1.

    34. Find the quotient of 12x2+x11 and 3x2.

Answers

  1. x2x+1

  2. y33y2+y+5

  3. x45x3x2+6x+2

  4. Trinomial; degree 2

  5. Trinomial; degree 4

  6. Binomial; degree 4

  7. Quadratic, −9

  8. Linear, 2

  9. Quadratic, 5

  10. Quadratic, −2

  1. 7x29x+5

  2. 2x2+5x

  3. 9x2+9x3

  4. 12x274x12

  5. x2y2+2xy1

  6. 2a2b2+12ab8

  7. 9x2y+6xy2

  8. 7x2y23xy6

  9. 32a258ab120b2

  10. 3x2n+2xn3

  11. y2+3y7

  12. SA=4πh2

  1. 16x3

  2. 10x22x

  3. 14x342x2

  4. 5y6+10y515y4

  5. 2x3y14x2y2+2xy3

  6. xn+2+xn+1+xn

  7. x2x20

  8. 6x211x+3

  9. 3x416x2y2+5y4

  10. 9x225

  11. a4b4

  12. 12x315x2y24xy+5y3

  13. x38x2+23x40

  14. 2x4+11x324x24x+1

  15. x2+16x+64

  16. 4x220x+25

  17. a26ab+9b2

  18. x4+4x2y2+4y4

  19. a42a3+11a10a+25

  20. x39x2+27x27

  21. 27x3+27x2+9x+1

  22. x4+8x3+24x2+32x+16

  23. 16x432x3+24x28x+1

  24. x4n25

  25. x4n2x2n+1

  26. 3x317x2+4x+4

  27. 9x6 square units

  1. 5x

  2. 5x23x+1

  3. 3a+7b1

  4. x22x+3

  5. 3x2x+2

  6. 4x2+5x614x3

  7. 2x3+3x25x+3

  8. 10x3+x2+4x+3+22x+1

  9. 3x3+6x2+13x+26+51x2

  10. x2+2x+42x2

  11. y4y3+y2y+1

  12. x23x+13x2x+2
  13. 2x3

  14. x2+4x1

  15. x2+5xyy2

  16. 4a2+2ab+b2

  17. 5x3