Expand and simplify 𝑎 minus five 𝑏 multiplied by 𝑎 plus five 𝑏 plus 24𝑎𝑏, then factorize the result completely.
We can begin this question by expanding the brackets, otherwise known as distributing the parentheses. One way of doing this is using the FOIL method. We begin by multiplying the first terms. 𝑎 multiplied by 𝑎 is equal to 𝑎 squared. Multiplying the outside terms gives us five 𝑎𝑏. Multiplying the inner terms gives us negative five 𝑎𝑏. Finally, multiplying the last terms gives us negative 25𝑏 squared. The middle two terms cancel as five 𝑎𝑏 minus five 𝑎𝑏 is equal to zero. Our expression simplifies to 𝑎 squared minus 25𝑏 squared.
An alternative method here would be to have recognized that we had an expression of the form the difference of two squares, 𝑥 minus 𝑦 multiplied by 𝑥 plus 𝑦. We know that an expression of this type is equal to 𝑥 squared minus 𝑦 squared. In this question, squaring 𝑎 gives us 𝑎 squared and squaring five 𝑏 gives us 25𝑏 squared. This confirms that our expression was correct. Our original expression is therefore equal to 𝑎 squared minus 25𝑏 squared plus 24𝑎𝑏.
We can switch the order of the last two terms so that this is equal to 𝑎 squared plus 24𝑎𝑏 minus 25𝑏 squared. We need to factor or factorize this into two sets of parentheses. The first term in both of these must be equal to 𝑎 as 𝑎 multiplied by 𝑎 is 𝑎 squared. The last terms need to multiply to give us negative 25𝑏 squared. We have three options here: negative five 𝑏 and five 𝑏, negative 25𝑏 and one 𝑏, or negative 𝑏 and 25𝑏. The middle term is equal to 24𝑎𝑏, which could be rewritten as 24𝑏𝑎.
This means that the correct pair must sum to give us 24𝑏. Negative five 𝑏 plus five 𝑏 is equal to zero, negative 25𝑏 plus 𝑏 is negative 24𝑏, and negative 𝑏 plus 25𝑏 is 24𝑏. This means that the third option is correct. The two parentheses are 𝑎 minus 𝑏 and 𝑎 plus 25𝑏. We can check this answer by redistributing the parentheses and collecting like terms. This would give us 𝑎 squared plus 24𝑎𝑏 minus 25𝑏 squared. The expression 𝑎 minus five 𝑏 multiplied by 𝑎 plus five 𝑏 plus 24𝑎𝑏 is equal to 𝑎 minus 𝑏 multiplied by 𝑎 plus 25𝑏.