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Polynomials and Radicals Help (page 2)

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Updated on Oct 27, 2011

Three Types of Factoring

There are three basic types of factoring:

1.  Factoring out a common monomial:

18x2 – 9x = 9x(2x – 1) abcb = b(ac)

2.  Factoring a quadratic trinomial using FOIL in reverse 

x2x – 20 = (x – 5)(x + 4)

x2 – 6x + 9 = (x – 3)(x – 3) = (x – 3)2

(for more on quadratic trinomials, see Quadratic Trinomials, Quadratic Equations, and the Quadratic Formula Help):

3.  Factoring the difference between two perfect squares using the rule a2b2 = (a + b)(ab):

x2 – 81 = (x + 9)(x – 9)

x2 – 49 = (x + 7)(x – 7)

Using Common Factors

With some polynomials, you can determine a common factor for all of the terms. For example, 4x is a common factor of all three terms in the polynomial 16x4 + 8x2 + 24x because it can divide evenly into each of them. To factor a polynomial with terms that have common factors, you can divide the polynomial by the known factor to determine the second factor.

Example

In the binomial 64x3 + 24x, 8x is the greatest common factor of Both terms. Therefore, you can divide 64x3 + 24x by 8x to find the Other factor.

= 8x2 + 3

Thus, factoring 64x3 + 24x results in 8x(8x2 + 3).

Fractions, Variables, and Factoring

Isolating Variables Using Fractions

It may be necessary to use factoring in order to isolate a variable in an equation.

Example

If axc = bx + d, what is x in terms of a, b, c, and d?

First isolate the x terms on the same side of the equation:

axbx = c + d

Now factor out the common x term:

x(ab) = c + d

Then divide both sides by ab to isolate the variable x:

Simplify:

x =

Fractions with Variables

You can work with fractions that contain variables in the same way as you would work with fractions without variables.

Example

Write as a single fraction.

First determine the LCD of 6 and 12: The LCD is 12. Then convert each fraction into an equivalent fraction with 12 as the denominator:

Then simplify:

Reciprocal Rules

There are special rules for the sum and difference of reciprocals. The following formulas can be memorized to save time when answering questions about reciprocals:

  • If x and y are not 0, then + y = x +
  • If x and y are not 0, then = y

Finding Roots

The roots of an equation are the values that make the equation true. For example, what are the roots of x3 – 9x2 – 10x = 0?

First, factor out the variable x: x(x2 – 9x – 10). Then, factor again: x2 – 9x – 10 = (x – 10)(x + 1), and x3 – 9x2 – 10x = x(x – 10)(x + 1). Set each factor equal to 0 and solve for x: x = 0; x – 10 = 0, x = 10; x + 1 = 0, x = –1. The roots of this equation are 0, –1, and 10.

Undefined Expressions

A fraction is undefined when its denominator is equal to 0. If the denominator of a fraction is a polynomial, factor it and set the factors equal to 0. The values that make the polynomial equal to 0 are the values that make the fraction undefined.

For what values of x is the fraction undefined?

Factor the denominator and set each factor equal to 0 to find the values of x that make the fraction undefined: x3 + 7x2 = x2(x + 7), x2 = 0, x = 0; x + 7 = 0, x = –7. The fraction is undefined when x equals 0 or –7.

Radicals

In order to simplify some expressions and solve some equations, you will need to find the square or cube root of a number or variable. The radical symbol, √ , signifies the root of a value. The square root, or second root, of x is equal to √x, or 2x. If there is no root number given, it is assumed that the radical symbol represents the square root of the number. The number under the radical symbol is called the radicand.

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