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Home >> Polynomials >> Multiplication of Polynomials >> Multiplication of Two or more Monomials >>

Multiplication of Two or more Monomials

Multiplication of Two or more Monomials Multiplication of Two or more Binomials Multiplication of Two or more Trinomials Multiplication of Monomial & Binomial Multiplication of Monomial & Trinomial
Multiplication of Binomial & Trinomial

Before you understand this concept, you are advised to read:

What are Monomials ?
How to Multiply Integers ?
What are like terms ?
What are Unlike terms ?

While multiplying two monomials you will find the following situations:

  • Multiply two monomials having like terms
    Example 1: Multiply (5a) and (10a)

  • Multiply two monomials having unlike terms
    Example 2: Multiply (3p) and (7q)

  • Multiply two negative monomials and both have like terms
    Example 3: Multiply (-4d) and (-2d)

  • Multiply two negative monomials having unlike terms
    Example 4: Multiply (-8y) and (-7z)

  • Multiply two monomials where one monomial is positive & other is negative and both have like terms
    Example 5: Multiply (-6s) and (2s)

  • Multiply two monomials where one monomial is positive & other is negative and have unlike terms
    Example 6: Multiply (-3p) and (7q)


    Multiply two monomials having like terms

    Example 1: Multiply 5a and 10a
    Solution: This proceeds as:
    Write terms in the multiplication expression and we get:
    5a X 10a

    Multiply constants and variable separately, as shown below:
    (5 X 10 ) X (a X a)

    Multiply constants as we multiply numbers & we get:
    (50) X (a X a)

    Multiply variables as per the Law of exponent, am X an = am+n, we get:
    50 X a 2

    Multiply constant and variable and we get:
    50a2

    Hence, (5a X 10a) = (50a2)


    Multiply two monomials having unlike terms

    Example 2: Multiply (3q) and (7p)
    Solution: This proceeds as:
    Write terms in the multiplication expression and we get:
    3q X 7p

    Multiply constants and variable separately, as shown below:
    (3 X 7 ) X (q X p)

    Multiply constants as we multiply numbers & we get:
    (21) X (q X p)

    Multiply variables & we get:
    21 X qp

    Multiply constant and variables and we get:
    21qp

    Or we write it as: 21pq (because variables are arranged in alphabetical order)

    Hence, (3p X 7q) = (21pq)


    Multiply two negative monomials and both have like terms

    Example 3: Multiply (-4d) and (-2d)
    Solution: This proceeds as:
    Write terms in the multiplication expression and we get:
    -4d X -2d

    Multiply constants and variable separately, as shown below:
    (-4 X -2 ) X (d X d)

    Multiply constants as we multiply negative integers & we get:
    (8) X (d X d)

    Multiply variables as per the Law of exponent, am X an = am+n, we get:
    8 X d2

    Multiply constant and variables and we get:
    8d2

    Hence, (-4d X -2d) = (8d2)


    Multiply two negative monomials having unlike terms

    Example 4: Multiply (-8y) and (-7z)
    Solution: This proceeds as:
    Write terms in the multiplication expression and we get:
    -8y X -7z

    Multiply constants and variable separately, as shown below:
    (-8 X -7) X (y X z)

    Multiply constants as we multiply negative integers & we get:
    56 X (y X z)

    Multiply variables & we get:
    56 X yz

    Multiply constant and variables and we get:
    56yz

    Hence, (-8y X -7z) = (56yz)


    Multiply two monomials where one monomial is positive & other is negative and both have like terms

    Example 5: Multiply (-6s) and (2s)
    Solution: This proceeds as:
    Write terms in the multiplication expression and we get:
    -6s X 2s

    Multiply constants and variable separately, as shown below:
    (-6 X 2 ) X (s X s)

    Multiply constants as we multiply positive and negative integers & we get:
    (-12) X (s X s)

    Multiply variables as per the Law of exponent, am X an = am+n, we get:
    -12 X s2

    Multiply constant and variables and we get:
    (-12s2)

    Hence, (-6s X 2s) = (-12s2)

    Multiply two monomials where one monomial is positive & other is negative and have unlike terms

    Example 6: Multiply (-3p) and 7q
    Solution: This proceeds as:

    Write terms in the multiplication expression and we get:
    -3p X 7q

    Multiply constants and variable separately, as shown below:
    (-3 X 7 ) X (p X q)

    Multiply constants as we multiply positive and negative integers & we get:
    (-21) X (p X q)

    Multiply variables & we get:
    -21 X pq

    Multiply constant and variables and we get:
    -21pq

    Hence, (-3p X 7q) = (-21pq)

    By now you must have understood how to multiply two monomials under different situations, below you can find some more examples where more than two monomials are multiplied:

    Example 7: Multiply (2p)(-3q)(2p)
    Solution: This proceeds as:
    Write terms in the multiplication expression and we get:
    2p X -3q X 2p

    Multiply constants and variable separately, as shown below:
    (2 X -3 X 2) X (p X q X p)

    Multiply constants as we multiply integers & we get:
    (-12) X (p X q X p)

    Rearrange variable & we get:
    (-12) X (p X p X q)

    Multiply Variables and we get:
    -12 X p2q

    Multiply constant and variables and we get:
    -12p2q

    Hence, (2p X -3q X 2p) = (-12p2q)

    Example 8: Multiply (-x)(-3xy)(6y)
    Solution: This proceeds as:
    Write terms in the multiplication expression and we get:
    -x X-3xy X 6y

    Multiply constants and variable separately, as shown below:
    (-1 X -3 X 6) X (x X xy X y)

    Multiply constants as we multiply integers & we get:
    (18) X (x X xy X y)

    Rearrange variable & we get:
    (18) X (x X x X y X y)

    Multiply Variables and we get:
    18 X x2y2

    Multiply constant and variables and we get:
    18 X x2y2

    Hence, (-x X-3xy X 6y) = (18 X x2y2)

    Example 9: Multiply (-2a)(-3c)(-3d)(4b)
    Solution: This proceeds as:
    Write terms in the multiplication expression and we get:
    -2a X -3c X -3d X 4b

    Multiply constants and variable separately, as shown below:
    (-2 X -3 X -3 X 4) X (a X c X d X b)

    Multiply constants as we multiply integers & we get:
    (-72) X (a X c X d X b)

    Multiply Variables and we get:
    (-72) X acdb

    Multiply constant and variables and we get:
    -72acdb

    Or we write it as: -72abcd (because variables are arranged in alphabetical order)

    Hence, (-2a X -3c X -3d X 4b) = (-72abcd)

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