Polynomials and Radicals Help (page 2)
Introduction to Polynomials
A monomial is an expression that is a number, a variable, or a product of a number and one or more variables.
6 y 5xy2 19a6b4
A polynomial is a monomial or the sum or difference of two or more monomials.
7y5 –6ab4 8x + y3 8x + 9y – z
Kinds of Polynomials
Here are three kinds of polynomials:
- A monomial is a polynomial with one term, such as 5b6.
- A binomial is a polynomial with two unlike terms, such as 2x + 4y.
- A trinomial is a polynomial with three unlike terms, such as y3 + 8z – 2.
Operations With Polynomials
To add polynomials, simply combine like terms. Let's look at (5y3 – 2y + 1) + (y3 + 7y – 4).
First remove the parentheses: 5y3 – 2y + 1 + y3 + 7y – 4. Then arrange the terms so that like terms are grouped together: 5y3 + y3 – 2y + 7y + 1 – 4. Now combine like terms: 6y3 + 5y – 3. Just to make sure you've got it, let's look at another example.
(2x – 5y + 8z) – (16x + 4y – 10z)
First remove the parentheses. Be sure to distribute the subtraction correctly to all terms in the second set of parentheses:
2x – 5y + 8z – 16x – 4y + 10z
Then arrange the terms so that like terms are grouped together:
2x – 16x – 5y – 4y + 8z + 10z
Now combine like terms:
–14x – 9y + 18z
To multiply monomials, multiply their coefficients and multiply like variables by adding their exponents.
(–4a3b)(6a2b3) = (–4)(6)(a3)(a2)(b)(b3) = –24a5b4
To divide monomials, divide their coefficients and divide like variables by subtracting their exponents.
To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and add the products.
8x(12x – 3y + 9)
Distribute. (8x)(12x) – (8x)(3y) + (8x)(9) Simplify. 96x2 – 24xy + 72x
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and add the quotients.
= x – 3y + 7
The FOIL method is used when multiplying binomials. FOIL represents the order used to multiply the terms: First, Outer, Inner, and Last. To multiply binomials, you multiply according to the FOIL order and then add the products.
(4x + 2)(9x + 8)
F: 4x and 9x are the first pair of terms.
O: 4x and 8 are the outer pair of terms.
I: 2 and 9x are the inner pair of terms.
L: 2 and 8 are the last pair of terms.
Now multiply according to FOIL:
(4x)(9x) + (4x)(8) + (2)(9x) + (2)(8) = 36x2 + 32x + 18x + 16
Now combine like terms:
36x2 + 50x + 16
Figuring Out Factoring
Introduction to Factoring
Factoring is the reverse of multiplication. When multiplying, you find the product of factors. When factoring, you find the factors of a product.
Multiplication: 3(x + y) = 3x + 3y
Factoring: 3x + 3y = 3(x + y)
Three Types of Factoring
There are three basic types of factoring:
1. Factoring out a common monomial:
18x2 – 9x = 9x(2x – 1) ab – cb = b(a – c)
2. Factoring a quadratic trinomial using FOIL in reverse
x2 – x – 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 a2 – b2 = (a + b)(a – b):
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 + 8
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.
If ax – c = 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:
ax – bx = c + d
Now factor out the common x term:
x(a – b) = c + d
Then divide both sides by a – b to isolate the variable x:
Fractions with Variables
You can work with fractions that contain variables in the same way as you would work with fractions without variables.
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:
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 –
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.
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.
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 2√x. 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.