Key Algebraic Identities
The square of a plus b equals a squared plus 2ab plus b squared. The square of a minus b equals a squared minus 2ab plus b squared. A squared minus b squared equals a plus b times a minus b. The square of a plus b minus the square of a minus b equals 4ab. The square of a plus b plus c equals a squared plus b squared plus c squared plus twice the sum of ab, bc and ca.
For cubes, a cubed plus b cubed equals a plus b times a squared minus ab plus b squared, and a cubed minus b cubed equals a minus b times a squared plus ab plus b squared. The cube of a plus b equals a cubed plus b cubed plus 3ab times a plus b. If a plus b plus c equals zero, then a cubed plus b cubed plus c cubed equals 3abc.
If x plus one over x equals k, then x squared plus one over x squared equals k squared minus 2, and x cubed plus one over x cubed equals k cubed minus 3k.
Factorization Methods
Take out a common factor. Group terms. Split the middle term: to factor x squared plus px plus q, find two numbers whose product is q and whose sum is p. For a x squared plus b x plus c, split b into two numbers with product a times c. The factor theorem says that if f of a equals zero then x minus a is a factor. The remainder theorem says that dividing f of x by x minus a leaves the remainder f of a.
Shortcuts
For large squares, write the number near a round figure. For example, 998 squared equals 1000 minus 2 all squared, which is 996004. A difference of squares such as 103 squared minus 97 squared equals 200 times 6, which is 1200. When the sum and product are given, find a squared plus b squared as the sum squared minus twice the product.
Worked Examples
Example 1: If x plus one over x equals 3, find x squared plus one over x squared.
Solution: Nine minus two equals 7.
Example 2: Factor x squared minus 5x plus 6.
Solution: Numbers with product 6 and sum minus 5 are minus 2 and minus 3, so the answer is x minus 2 times x minus 3.
Example 3: Evaluate 998 squared.
Solution: One thousand minus two, squared, equals 1,000,000 minus 4,000 plus 4, which is 996,004.
Quick Check
Question 1: What does a cubed plus b cubed plus c cubed equal when a plus b plus c is zero?
Answer: Three times a times b times c.
Question 2: State the factor theorem.
Answer: If f of a equals zero then x minus a is a factor.
Question 3: What is a squared minus b squared?
Answer: a plus b times a minus b.