Video Transcript
Expand 𝑚 plus four squared.
So, we have a binomial, 𝑚 plus four. And the first really important thing to
notice is that we are squaring this entire binomial. This means that we’re taking the
expression 𝑚 plus four and multiplying it by itself. So, another way of writing this would be
as 𝑚 plus four multiplied by 𝑚 plus four. The word expand is another way of saying
distribute. So, we want to expand the brackets or
distribute the parentheses. We need to multiply both terms in the
first binomial by both terms in the second.
Let’s look at a physical interpretation
of this first using area. Suppose we have a length of 𝑚 plus four
units. We can break this down into a length of
𝑚 units and a length of four units. When we multiply 𝑚 plus four by itself,
this is equivalent to finding an expression for the area of a square with side lengths of 𝑚
plus four units. We can divide this area up into four
smaller regions and find each of their individual areas. They’re each either rectangles or
squares, so we find their areas by multiplying their dimensions together.
The first region has an area of 𝑚
multiplied by 𝑚 which is 𝑚 squared. The second region has an area of 𝑚
multiplied by four, which is four 𝑚. The third region has an area of four
multiplied by 𝑚, which is another lot of four 𝑚. And the final region has an area of four
multiplied by four, which is 16. The total area can be found by summing
the four individual areas, giving 𝑚 squared plus four 𝑚 plus four 𝑚 plus 16. Now, we must remember to simplify this
expression by grouping like terms. And the only like terms are those in the
center of our expansion, plus four 𝑚 plus four 𝑚, which makes positive eight 𝑚. Our expression, therefore, simplifies to
𝑚 squared plus eight 𝑚 plus 16. And this is the expanded form of 𝑚 plus
four all squared.
Now, notice that, at this stage here, we
have four terms in our expansion, and the middle two are exactly the same. Once we’ve grouped the like terms, we
have three terms in our expansion. This will always be the case when
squaring a binomial. So, if you find you have a different
number of terms or you don’t have two terms which are exactly the same, then something is
gone wrong. So, you need to check your work. So, that’s one approach to consider this
expansion as the area of a square with side lengths of 𝑚 plus four units. This is also sometimes referred to as a
grid method because it follows the same principles as the grid method for
multiplication.
A second method we could consider would
be to use the distributive property of multiplication. If we’re multiplying 𝑚 plus four by 𝑚
plus four, then we can write this as 𝑚 multiplied by 𝑚 plus four plus four multiplied by
𝑚 plus four. So, we’ve distributed the terms in our
first binomial over the second. We then have to multiply out or expand or
distribute each of these single sets of parentheses. 𝑚 multiplied by 𝑚 gives 𝑚 squared. 𝑚 multiplied by four gives four 𝑚. We then have four multiplied by 𝑚, which
gives another lot of four 𝑚, and finally four multiplied by four, which gives 16.
Our expression is now identical to the
one that we had at this stage here in our first method. And the only remaining step is to group
the like terms in the center of our expansion. As before, we find that 𝑚 plus four all
squared is equal to 𝑚 squared plus eight 𝑚 plus 16. So, that’s two possible methods, but
there is a third. And this one is perhaps the most
popular. It’s called the FOIL method. And the word FOIL is an acronym, meaning
that each of its letters stands for something. It’s just a way of ensuring that we
multiply all of the correct pairs of terms together, and we don’t miss any out.
The letter F in the word FOIL stands for
first. So, this means we multiply the first term
in the first binomial by the first term in the second. That’s 𝑚 multiplied by 𝑚, which is, of
course, 𝑚 squared. The letter O stands for outers or
outside. So, we multiply the terms that are on the
outside of our expansion. That’s the 𝑚 in the first binomial and
the four in the second, 𝑚 multiplied by four, which is four 𝑚. You’ve probably guessed that the I stands
for inners or inside, so we multiply together the terms in the inside of the expansion. That’s the four in the first binomial and
the 𝑚 in the second, four multiplied by 𝑚, which gives four 𝑚. And finally, L, which stands for
lasts. We multiply together the last term in
each binomial. That’s the four in the first binomial and
the four in the second, four multiplied by four, which is 16.
As before, we should always have four
terms in our expansion at this point. And we have two like terms in the center
of the expansion, which can be combined. Using all three methods then, we’ve
arrived at the same result. When we expand 𝑚 plus four all squared,
which means to multiply the binomial 𝑚 plus four by itself, we get the answer 𝑚 squared
plus eight 𝑚 plus 16. It is usual, although not essential, to
write the terms in our expansion in this order, that is, decreasing powers of 𝑚. So, we have our 𝑚 squared term first,
then our 𝑚 term, and then finally the constant term.
