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Algebra 1

6.2.3 Multiplying a Polynomial by a Polynomial

Algebra 16.2.3 Multiplying a Polynomial by a Polynomial
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6.2.3 • Multiplying a Polynomial by a Polynomial

Activity

For questions 1 – 5, match each polynomial expression to its simplified solution. Be prepared to explain or show your work.

1. ( c + 1 ) ( c 2 + 3 c + 6 ) ( c + 1 ) ( c 2 + 3 c + 6 )

2. ( c − 6 ) ( c 2 + c + 1 ) ( c − 6 ) ( c 2 + c + 1 )

3. ( c + 5 ) ( c 2 + 2 c − 4 ) ( c + 5 ) ( c 2 + 2 c − 4 )

4. ( c 2 − 3 ) ( c 2 − 4 c − 1 ) ( c 2 − 3 ) ( c 2 − 4 c − 1 )

5. ( c 2 + 10 ) ( c 2 − 5 c + 2 ) ( c 2 + 10 ) ( c 2 − 5 c + 2 )

6. Now, fill in the missing blanks to complete the polynomial multiplication. ( 2 x − 4 ) ( x 2 + ◻ x − 5 ) = 2 x ( x 2 + 6 x − 5 ) − 4 ( x 2 + 6 x − ◻ ) = 2 x 3 + ◻ x 2 − 10 x − 4 x 2 − 24 x + ◻ = ◻ x 3 + 8 x 2 − ◻ x + 20 ( 2 x − 4 ) ( x 2 + ◻ x − 5 ) = 2 x ( x 2 + 6 x − 5 ) − 4 ( x 2 + 6 x − ◻ ) = 2 x 3 + ◻ x 2 − 10 x − 4 x 2 − 24 x + ◻ = ◻ x 3 + 8 x 2 − ◻ x + 20

Video: Multiplying Polynomials Using Different Methods

Watch the following video to learn more about how to multiply polynomials using both the Distributive Property and Vertical Alignment.

Access multimedia content

Self Check

Multiply.

( d + 4 ) ( 2 d 2 − 3 d + 5 )

  1. 8 d 2 − 12 d + 20
  2. 2 d 3 − 3 d 2 + 5 d
  3. 2 d 3 + 5 d 2 − 7 d + 20
  4. 2 d 3 − 11 d 2 − 17 d + 20

Additional Resources

Multiplying a Polynomial by a Polynomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use other representations of the Distributive Property such as Vertical Alignment or distributing individual terms or polynomials.

Example 1

Multiply ( b + 3 ) ( 2 b 2 − 5 b + 8 ) ( b + 3 ) ( 2 b 2 − 5 b + 8 ) using the Distributive Property by distributing either a binomial or an individual term.

Step 1 - Distribute.

( 2 b 2 − 5 b + 8 ) ( 2 b 2 − 5 b + 8 ) .

b ( 2 b 2 − 5 b + 8 ) + 3 ( 2 b 2 − 5 b + 8 ) b ( 2 b 2 − 5 b + 8 ) + 3 ( 2 b 2 − 5 b + 8 )

Step 2 - Multiply.

2 b 3 − 5 b 2 + 8 b + 6 b 2 − 15 b + 24 2 b 3 − 5 b 2 + 8 b + 6 b 2 − 15 b + 24

Step 3 - Combine like terms.

2 b 3 + b 2 − 7 b + 24 2 b 3 + b 2 − 7 b + 24

Example 2

Multiply ( b + 3 ) ( 2 b 2 − 5 b + 8 ) ( b + 3 ) ( 2 b 2 − 5 b + 8 ) using vertical alignment.

It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Step 1 - Multiply.

( 2 b 2 − 5 b + 8 ) ( 2 b 2 − 5 b + 8 ) by 3 3 .

Step 2 - Multiply.

( 2 b 2 − 5 b + 8 ) ( 2 b 2 − 5 b + 8 ) by b b .

Step 3 - Add like terms.

2 b 2 − 5 b + 8 × b + 3 ― 6 b 2 − 15 b + 24 2 b 3 − 5 b 2 + 8 b ― 2 b 3 + b 2 − 7 b + 24 2 b 2 − 5 b + 8 × b + 3 ― 6 b 2 − 15 b + 24 2 b 3 − 5 b 2 + 8 b ― 2 b 3 + b 2 − 7 b + 24

Try it

Multiplying a Polynomial by a Polynomial

For questions 1 – 2, multiply ( z − 3 ) ( z 2 − 5 z + 2 ) ( z − 3 ) ( z 2 − 5 z + 2 ) using the listed method.

1. The Distributive Property

2. Vertical Alignment

Write down your answer, then select the solution button to compare your work.

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