Video Transcript
Expand the product 𝑥 plus four
multiplied by 𝑥 plus six.
Here, we’re asked to expand the
product of two algebraic expressions, 𝑥 plus four and 𝑥 plus six. In fact, these expressions are both
binomials because they each contain two terms. When multiplying two binomials
together, we must make sure we multiply each term in the first binomial by each term
in the second, which will give four terms overall, before we perform any
simplification.
We can do this in a number of
ways. One way is to use the grid
method. We can picture the product of these
two expressions as a rectangle with side lengths of 𝑥 plus four and 𝑥 plus six
units. The product of these expressions
corresponds to the area of the rectangle, which we divide into four smaller
rectangles.
We can find the area of each of
these rectangles by multiplying their length by their width. For the top-left rectangle, 𝑥
multiplied by 𝑥 gives 𝑥 squared. Then, for the bottom-left
rectangle, 𝑥 multiplied by six is six 𝑥. The final two rectangles have areas
of four 𝑥 and 24 square units. Summing these four expressions
gives an expression for the total area of the larger rectangle and, hence, an
expression for the given product. Finally, collecting like terms
gives that the product of 𝑥 plus four and 𝑥 plus six is equal to 𝑥 squared plus
10𝑥 plus 24.
An alternative method is just to
use a systematic approach to ensure we multiply each term in the first binomial by
each term in the second. If we multiply the first terms in
each binomial together, we obtain 𝑥 squared. Then, if we multiply the terms on
the outside of the product together, we obtain six 𝑥. Multiplying the terms on the inside
of the product together gives four 𝑥. And finally multiplying the last
term in each binomial together gives 24.
We obtained four terms as expected,
and then we can simplify by grouping the like terms. We’ve found that the expanded and
simplified form of the product 𝑥 plus four multiplied by 𝑥 plus six is 𝑥 squared
plus 10𝑥 plus 24.
