Factoring into Special Products

Multiply the following polynomials.

(𝑥 + 3)(𝑥! − 3𝑥 + 9) (2𝑥 − 5𝑦)(4𝑥! + 10𝑥𝑦 + 25𝑦!)

𝑥!          − 3𝑥          + 9 4𝑥!  + 10𝑥𝑦  + 25𝑦!

𝑥 2𝑥

+3 −5𝑦

Product: _________________ Product: _________________

What do you notice is special about each of the terms in the final products?

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How are each of the terms in the final products related to the terms in the binomial factors?

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How are each of the terms in the binomial factors related to the terms in the trinomial factors?

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Using the patterns seen above, factor the following binomials (referred to as a sum of cubes and difference of cubes).

𝑥! + 1000 27ℎ! − 216𝑝!

                  !   +                       !                   !   −                     !

(                  +                  )(                    −                    +                    ) (                −                  )(                  +                    +                    )

We can now determine the formulas for factoring expressions that look like a sum or difference of two cubes. Allow the first term being cubed to be “F” and the last term being cubed to be “L”.

𝑭𝒊𝒓𝒔𝒕 𝟑   +     𝑳𝒂𝒔𝒕 𝟑 𝑭𝒊𝒓𝒔𝒕 𝟑   −   𝑳𝒂𝒔𝒕 𝟑

(                                        )(                                                                    ) (                                        )(                                                                    )

Multiply the following polynomials.

(𝑥 + 4)(𝑥 − 4) (3𝑥 − 7𝑦)(3𝑥 + 7𝑦)

𝑥                              − 4              3𝑥                      + 7𝑦

𝑥 3𝑥

+4 −7𝑦

Product: _________________ Product: _________________

What do you notice is special about each of the terms in the final products?

______________________________________________________________________________

______________________________________________________________________________

How are each of the terms in the final products related to the terms in the binomial factors?

______________________________________________________________________________

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What do you notice about the signs in the binomial factors compared to the signs in the products?

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Using the patterns seen above, factor the following binomials (referred to as a difference of squares).

𝑥! − 144 25𝑎! − 81𝑏!

                  !   −                       !                   !    −                     !

(                  +                  )  (                −                  ) (                  +                  )  (                −                  )

We can now determine the formula for factoring expressions that look like a difference of two squares. Allow the first term being squared to be “F” and the last term being squared to be “L”.

𝑭𝒊𝒓𝒔𝒕 𝟐   −   𝑳𝒂𝒔𝒕 𝟐

(                                        )  (                                        )