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Published byKaitlyn Fisher Modified over 3 years ago

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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 the problem 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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23 x____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 the 20 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. 6x 2 y + 3xy 2

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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: 7x 2 (2xy – 3x 2 ) 2xy – 3x 2 7x 2 x ________ Align the terms vertically with the monomial under the polynomial. Now multiply each term in the polynomial by the monomial. – 21x 4 14x 3 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)(x – 2) Remember that we could use a vertical format when multiplying a polynomial by monomial. We can do the same here. (x + 3) (x – 2) x ________ Line up the terms by degree. Multiply in the same way you would multiply two 2- digit numbers. – 62x + 0+ 3xx2x2 _________ – 6+ 5xx2x2

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Multiply: (x + 3)(x – 2) (x + 3) (x – 2) x ________ – 62x + 0+ 3xx2x2 _________ – 6+ 5xx2x2 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. (x 2 – 3x + 2)(x 2 – 3) (x 2 – 3x + 2) (x 2 – 3) x ____________ Line up like terms. – 6+ 9x– 3x 2 + 0+ 0x+ 2x 2 – 3x 3 x4x4 __________________ – 6+ 9x– 1x 2 – 3x 3 x4x4

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It is also advantageous to multiply polynomials without rewriting them in a vertical format. Multiply: (x + 2)(x – 5) 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 (x – 5).

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(x + 2)(x – 5) 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: (x – 6)(2x + 1) x(2x)+ x(1)– (6)2x– 6(1) 2x 2 + x – 12x – 6 2x 2 – 11x – 6

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1. 2x 2 (3xy + 7x – 2y) 2. (x + 4)(x – 3) 3. (2y – 3x)(y – 2)

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2x 2 (3xy + 7x – 2y) 2x 2 (3xy) + 2x 2 (7x) + 2x 2 (–2y) 2x 2 (3xy + 7x – 2y) 6x 3 y + 14x 2 – 4x 2 y

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(x + 4)(x – 3) (x + 4)(x – 3) x(x) + x(–3) + 4(x) + 4(–3) x2 x2 – 3x + 4x – 12 x2 x2 + x –

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(2y – 3x)(y – 2) (2y – 3x)(y – 2) 2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2) 2y 2 – 4y – 3xy + 6x

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