remindr

A "polynomial" may contain one term (monomial), two terms (binomial), three terms (trinomial), or more terms.

We have seen monomials multiplied by other polynomials, and binomials multiplied by binomials.
Now, let's see what happens when other combinations of polynomials are multiplied.

hint gal
There are various set-up methods possible when multiplying polynomials. As the number of terms in the polynomials increases, the vertical multiplication set-up is much more likely to yield a correct result. It neatly organizes your work which helps eliminate careless errors.

The following examples will show various set-up methods for the multiplications.
Notice that FOIL will not work to solve any of these problems.

expin1
Multiply: (x - 2)(x² + 3x - 5)
Vertical Set-Up: Line up the polynomials as you would for numerical multiplication. Be careful of your signs.

polym2

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

polym1

Grid Set-Up:
• 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.

polygrid3

x³ + 3x² - 5x - 2x² - 6x + 10
= x³ + x² - 11x + 10

 

expin2
Multiply: (2x + 7)(x³ + 4x² - 2x + 6)
Horizontal Set-Up: Multiply each term of the binomial times each term of the cubic polynomial. There will be 8 multiplications. Combine the like terms.

plolym2H

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

polym2V

Grid Set-Up: Set up the grid and multiply.

GRID4

POLYM2G

 

expin3
Multiply: (x + 2)(x² - 4) (missing term in second factor)
Horizontal Set-Up: Multiply each term of the binomial times each term of the cubic polynomial. There will be 8 multiplications. Combine the like terms.

abc1

Vertical Set-Up: 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.

abc22

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

abc3

Combining the terms by adding along the diagonals:abc4


example4
Multiply: (2x² - x - 1)(x² - 4x - 2)
Horizontal Set-Up: Multiply each term of the first trinomial times each term of the second trinomial. There will be 9 multiplications. Combine the like terms.

polymm3H

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

poly3mV

Grid Set-Up: Set up the grid and multiply.

GRID5A

POLYM3G

 

 

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