Castle Learning Reference

Reference Tables for Pre-Calculus

Advanced Factoring Techniques

Grouping

Grouping is strategy of factoring polynomials and solving equations using greatest common factoring. Grouping will not work for all polynomials. Here is an example of when you can use grouping and how to do it.

Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a linear factor such as x – 1. It is often used to find the roots of a polynomial function.

Example: Divide 4x³ – 6x + 1 by (x + 1) or x = -1. Note that dividing by (x + 1) or x = -1 is equivalent. If set the factors in parentheses equal to 0 and solve for x, we get x = -1.

Step 1:

Place the root into the '

' and then list the coefficients in a row. Remember that 4x³ – 6x + 1 = 4x³ + 0x² – 6x + 1. Don't forget the 0 coefficient of x². Now put a line under it as shown above.

Step 2:

Drop the first coefficient under the line.

Step 3:

Multiply the number inside the '

' by the number under the line. Place the product under the second coefficient and then add down. Place the sum of the two numbers below the line as shown above.

Step 4:

Repeat step #3 with all of the other coefficients until completed.

When the division of complete, the first number in the bottom row becomes the new leading coefficient of x raised to one less than the degree of the original polynomial, in this case, 3 – 1 = 2. Then the second number is the coefficient of the next less power of x, and so on until the last number which has no positive exponent; this is the remainder. The remainder is written as the last number over the divisor, which in this case is (x + 1).

The result of

U-Substitution

U-substitution is a strategy of factoring and solving polynomial functions, using a "let" statement. U-substitution does not work for all polynomials. Here is an example of how to find the roots of a polynomial using U-substitution.