Expanding and Simplifying

Expanding means removing the brackets by multiplying. The number in front multiplies every term inside. Four times the bracket three x plus five is like a rectangle four tall and three x plus five wide. It splits into two pieces: four times three x is twelve x, and four times five is twenty. So the expansion is twelve x plus twenty.

When the term in front is negative, its sign goes into every multiplication. Negative three times the bracket two a minus four. Negative three times two a is negative six a. Negative three times negative four is positive twelve, because a negative times a negative is positive. So the answer is negative six a plus twelve.

When the term in front has a letter, the index laws come in. Two p times the bracket three p squared minus five p. Two p times three p squared: two times three is six, and p times p squared is p cubed, because you add the indices one and two. Two p times negative five p is negative ten p squared. The answer is six p cubed minus ten p squared.

Most questions have two brackets: expand each one, then collect like terms. Three x times the bracket two x minus five gives six x squared minus fifteen x. Then the second part is negative four times the bracket x minus seven. Negative four times x is negative four x, and negative four times negative seven is positive twenty-eight. Now collect: negative fifteen x and negative four x make negative nineteen x. The answer is six x squared minus nineteen x plus twenty-eight.

A minus sign in front of a bracket means negative one times the bracket. In five times the bracket two n plus one, minus the bracket n minus three, the first part is ten n plus five. The second part is negative one times n, which is negative n, and negative one times negative three, which is plus three. Collect to get nine n plus eight. Check by putting n equals two into both: the original gives twenty-five minus negative one, which is twenty-six, and nine times two plus eight is also twenty-six. Quiz and exercises now.