Question Video: Using an Area Model to Multiply Polynomials Mathematics

Use an area model to write an expression equivalent to (−4𝑎 + 𝑏)(3 − 2𝑏 − 𝑎).

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Video Transcript

Use an area model to write an expression equivalent to negative four 𝑎 plus 𝑏 multiplied by three minus two 𝑏 minus 𝑎. The expression negative four 𝑎 plus 𝑏 has been split into its two terms. Likewise, three minus two 𝑏 minus 𝑎 has been split into three terms. We can calculate the area of a rectangle by multiplying its length by its width. To calculate the area of the top-left rectangle, we need to multiply negative four 𝑎 by three. This is equal to negative 12𝑎.

The section next to this is equal to negative four 𝑎 multiplied by negative two 𝑏. Multiplying two negative terms gives a positive answer. Therefore, negative four 𝑎 multiplied by negative two 𝑏 is equal to eight 𝑎𝑏. The final section on the top row is equal to negative four 𝑎 multiplied by negative 𝑎. Once again, we have two negative terms, and 𝑎 multiplied by 𝑎 is 𝑎 squared. So negative four 𝑎 multiplied by negative 𝑎 is four 𝑎 squared.

We repeat this process for the bottom row. The first section has an area of three 𝑏. The middle section is equal to 𝑏 multiplied by negative two 𝑏. This is equal to negative two 𝑏 squared. Finally, we have 𝑏 multiplied by negative 𝑎. This is equal to negative 𝑎𝑏. The total area will be equal to the sum of these six terms. We can group or collect the like terms, eight 𝑎𝑏 minus 𝑎𝑏. This gives us seven 𝑎𝑏. Negative four 𝑎 plus 𝑏 multiplied by three minus two 𝑏 minus 𝑎 is equivalent to negative 12𝑎 plus seven 𝑎𝑏 plus four 𝑎 squared plus three 𝑏 minus two 𝑏 squared. These five terms can be written in any order.

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