Multiplication Involving Trinomials

Multiplication Involving Trinomials

Multiplying a Binomial and a Trinomial 

Put the “distributive method” of multiplication to work. Multiply each term from the binomial times each term of the trinomial. You can do this by using either a horizontal method or a vertical method.
Note: FOIL will not work in this problem.

Example 1: Multiply: (x – 2)(x² + 3x – 5)

Horizontal Method:
Multiply each term of the binomial times each term of the trinomial. There will be 6 multiplications. Combine the like terms.

Vertical Method:
Line up the polynomials as you would for numerical multiplication. Be careful of your signs.

Grid Method:

• Place one of the polynomials along the top and the other down the left side.
• Position the terms so that each term (and its sign) lines up with a row or column of the grid.
• Multiply each intersecting row and column to fill the interior of the grid.
• Copy and add all of the terms in the interior of the grid.
• Combine like terms.
  

Example 2: Multiply: (2x + 7)(x³ + 4x² – 2x + 6)
Horizontal Method: Multiply each term of the binomial times each term of the cubic polynomial. There will be 8 multiplications. Combine the like terms.

Vertical Method: Line up the polynomials as you would for numerical multiplication. Be careful of your signs.

Grid Method: Set up the grid and multiply.

Example 3: Multiply: (x + 2)(x² – 4) (missing term in second factor)
Horizontal Method: Multiply each term of the binomial times each term of the cubic polynomial. There will be 8 multiplications. Combine the like terms.

Vertical Method: Line up the polynomials as you would for numerical multiplication. Be careful of your signs. If a term is missing, replacing it with 0 (such as 0x in this example) may help you to line up the terms correctly.

Grid Method: Set up the grid and multiply.
Including the missing term will keep the diagonals working properly in the grid for adding purposes.

Combining the terms by adding along the diagonals:

Example 4: Multiply: (2x² – x – 1)(x² – 4x – 2)
Horizontal Method: Multiply each term of the first trinomial times each term of the second trinomial. There will be 9 multiplications. Combine the like terms.

Vertical Method: Line up the polynomials as you would for numerical multiplication. Be careful of your signs.

Grid Method: Set up the grid and multiply.

Cubing a Binomial

(multiplying the binomial times itself 3 times)

Method 1:
To cube a binomial, multiply it times itself three times. This will require a two step process.
STEP 1: Multiply the first two factors.

STEP 2: Multiply your answer by the third factor.

Method 2:
CASE 1:

Notice the pattern:
1. There are 4 terms in the pattern.
2. The exponents of a decrease in each term, while the exponents of b increase in each term.
3. The middle terms contain a factor of 3.

CASE: 2

The pattern is similar to CASE 1, but the signs of the second and fourth terms are negative.
Remember: if you forget the patterns, just multiply the three factors to get the answer.

Example:
Method 1:
(x + 1)³
= (x + 1)(x + 1)(x + 1)
= (x² + 2x + 1)(x + 1)
= x³ + 3x² + 3x + 1
Method 2:
(x + 1)³
= x³ + 3(x²)(1) + 3(x)(1²) + 13
= x³ + 3x² + 3x + 1

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