# Question Video: Expanding the Product of Two Binomials Mathematics • 9th Grade

Find 𝐴𝐵 given that 𝐴 = 3𝑎² + 5 and 𝐵 = 2𝑎² + 6.

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### Video Transcript

Find 𝐴𝐵 given that 𝐴 equals three 𝑎 squared plus five and 𝐵 equals two 𝑎 squared plus six.

We’ve been given two algebraic expressions, 𝐴 and 𝐵, and asked to find 𝐴𝐵, which is their product. We therefore need to find the product of three 𝑎 squared plus five and two 𝑎 squared plus six. Each of these algebraic expressions are binomials. And to find their product, we need to multiply out the brackets. In doing so, we need to ensure that we multiply each term in the first binomial by each term in the second. There are a number of different ways we can do this.

One method is the grid method, in which we represent the product as a rectangle with sides whose lengths are equal to the two factors. Because the area of a rectangle is equal to its length multiplied by its width, the area of the rectangle is equivalent to the product of the two factors. We can divide the rectangle into four smaller rectangles and find an expression for the area of each by multiplying their dimensions.

For the top-left rectangle, three 𝑎 squared multiplied by two 𝑎 squared is six 𝑎 to the fourth power. Remember that when we multiply powers of the same base together, we add the exponents. For the bottom-left rectangle, three 𝑎 squared multiplied by six is 18𝑎 squared. So far, we have multiplied the first term in binomial 𝐴 by each term in binomial 𝐵. Moving to the top right rectangle, five multiplied by two 𝑎 squared is 10𝑎 squared. And for the final rectangle, five multiplied by six is 30.

Finally, we find the total area of the rectangle by summing the expressions for the areas of the four smaller rectangles, giving six 𝑎 to the fourth power plus 18𝑎 squared plus 10𝑎 squared plus 30. Simplifying by combining the like terms in the center of the expression gives six 𝑎 to the fourth power plus 28𝑎 squared plus 30.

So, using the grid method, we’ve found that 𝐴𝐵, which is the product of the two binomials 𝐴 and 𝐵, is equal to six 𝑎 to the fourth power plus 28𝑎 squared plus 30.