9 5 Multiplying Polynomials

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    1. Multiply a polynomial by a monomial.

    2. Multiply a polynomial by a polynomial.

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    The Distributive Property

    Look at the following expression:

    3(x + 7) This expression is the sum of x and 7 multiplied by 3.

    To simplify this expression we can distribute the multiplication

    by 3 to each number in the sum.

    (3 x) + (3 7)

    3x + 21

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    Whenever we multiply two numbers, we are putting the distributive

    property to work.

    7(23) We can rewrite 23 as (20 + 3) then theproblem would look like 7(20 + 3).

    Using the distributive property:

    (7 20) + (7 3) = 140 + 21 = 161

    When we learn to multiply multi-digit

    numbers, we do the same thing in a vertical

    format.

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    23x____7

    7 3 = 21. Keep the 1 in the

    ones position then carry the 2

    into the tens position.

    1

    2

    7 2 = 14. Add the 2 from before

    and we get 16.16

    What weve really done in the second

    step, is multiply 7 by 20, then add the20 left over from the first step to get

    160. We add this to the 1 to get 161.

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    Multiply: 3xy(2x + y)

    This problem is just like the review problems except for a

    few more variables.

    To multiply we need to distribute the 3xy over the

    addition.

    3xy(2x + y) = (3xy 2x) + (3xy y) =

    Then use the order of operations and the properties of

    exponents to simplify.

    6x2y + 3xy2

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    We can also multiply a polynomial and a monomial using a vertical

    format in the same way we would multiply two numbers.

    Multiply: 7x2(2xy3x2)

    2xy3x2

    7x2x________

    Align the terms vertically with the

    monomial under the polynomial.

    Now multiply each term in thepolynomial by the monomial.

    21x2

    14x3

    y

    Keep track of negative

    signs.

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    To multiply a polynomial by another polynomial we use the

    distributive property as we did before.

    Multiply: (x + 3)(x2)

    Remember that we could use a vertical format when multiplying a

    polynomial by monomial. We can do the same here.

    (x + 3)(x2)x________

    Line up the terms by degree.

    Multiply in the same way

    you would multiply two 2-

    digit numbers.

    62x+ 0+ 3xx2_________

    6+ 5xx2

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    Multiply: (x + 3)(x2)(x + 3)(x2)x________

    62x+ 0+ 3xx2_________

    6+ 5xx2

    To multiply the problem below, we have distributed each term in one of

    the polynomials to each term in the other polynomial.

    Here is another example.

    (x23x + 2)(x23)

    (x23x + 2)

    (x2 3)x____________Line up like terms.

    6+ 9x3x2

    + 0+ 0x+ 2x23x3x4__________________6+ 9x1x23x3x4

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    It is also advantageous to multiply polynomials without rewriting

    them in a vertical format.

    Multiply: (x + 2)(x5)

    Though the format does not change, we must still distribute each

    term of one polynomial to each term of the other polynomial.

    Each term in (x+2) is distributed

    to each term in (x5).

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    (x + 2)(x5)

    This pattern for multiplying polynomials is called FOIL.

    Multiply the First terms.

    Multiply the Outside terms.

    Multiply the Inside terms.Multiply the Last terms.

    F

    O

    I

    L After you multiply, collect like

    terms.

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    Example: (x6)(2x + 1)

    x(2x) + x(1)(6)2x 6(1)

    2x2+ x12x6

    2x211x6

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    2x2(3xy + 7x2y)

    2x2(3xy) + 2x2(7x) + 2x2(2y)

    2x2(3xy + 7x2y)

    6x3y + 14x24x2y

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    (x + 4)(x3)

    (x + 4)(x3)

    x(x) + x(3) + 4(x) + 4(3)

    x23x + 4x12

    x2+ x12

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    (2y3x)(y2)

    (2y3x)(y2)

    2y(y) + 2y(2) + (3x)(y) + (3x)(2)

    2y24y3xy + 6x