Special Products

Squaring a bracket means multiplying it by itself. x plus four, all squared, is x plus four times x plus four. Draw the square: x squared in one corner, sixteen in the other, and two equal rectangles of four x. Those two rectangles are the middle term, eight x. So x plus four, all squared, is x squared plus eight x plus sixteen. The pattern is: square the first term, add twice the product, add the square of the last term.

For a difference, the pattern is the same but the middle term is negative. Two m minus three, all squared. Square the first term: two m squared is four m squared, because the two is squared too. Twice the product: two times two m times negative three is negative twelve m. Square the last term: negative three squared is positive nine. So the answer is four m squared minus twelve m plus nine.

When the two brackets have the same terms but one is plus and one is minus, something neat happens. x plus seven times x minus seven: x times x is x squared, x times negative seven is negative seven x, seven times x is positive seven x, and seven times negative seven is negative forty-nine. The two middle terms cancel, leaving x squared minus forty-nine. That is a difference of two squares.

Before you expand, decide which pattern you are looking at. If the same bracket appears twice, like p minus five all squared, it is a perfect square and the answer has three terms. If the brackets have the same two terms with opposite signs, like p plus five times p minus five, it is a difference of two squares and the answer has just two terms. Watch the order: five minus p times five plus p is twenty-five minus p squared, because five is the first term.

Special products turn hard multiplications into easy ones. Forty-nine times fifty-one is fifty minus one, times fifty plus one, which is a difference of two squares: fifty squared minus one squared, or two thousand five hundred minus one, which is two thousand four hundred and ninety-nine. Sixty-one squared is sixty plus one, all squared: sixty squared is three thousand six hundred, twice sixty times one is a hundred and twenty, and one squared is one. That makes three thousand seven hundred and twenty-one. Quiz and exercises now.