Polynomials and Radicals Help (page 4)
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.
Adding and Subtracting Radicals
Two radicals can be added or subtracted if they have the same radicand. To add two radicals with the same radicand, add the coefficients of the radicals and keep the radicand the same.
2√2 + 3√2 = 5√2
To subtract two radicals with the same radicand, subtract the coefficient of the second radical from the coefficient of the first radical and keep the radicand the same.
6√5 –4√5 = 2√5
The expressions √3 + √2 and √3 – √2 cannot be simplified any further, because these radicals have different radicands.
Two radicals can be multiplied whether they have the same radicand or not. To multiply two radicals, multiply the coefficients of the radicals and multiply the radicands.
(4√6)(3√7) =12√42, because (4)(3) =12 and (√6)(√7)=√42.
If two radicals of the same root with the same radicand are multiplied, the product is equal to the value of the radicand alone.
Here's an example: (√6)(√6) = 6. Both radicals represent the same root, the square root, and both radicals have the same radicand, 6, so the product of √6 and √6 is 6.
Two radicals can be divided whether they have the same radicand or not. To divide two radicals, divide the coefficients of the radicals and divide the radicands.
= 5√5, because = 5 and = √5.
Any radical divided by itself is equal to 1: = 1.
Simplifying a Single Radical
To simplify a radical such as √64, find the square root of 64. Look for a number that, when multiplied by itself, equals 64. Because (8)(8) = 64, the square root of 64 is 8: √64 = 8; √64 is expressed as 8, not –8. The equation x2 = 64 has two solutions, because both 8 and –8 square to 64, but the square root of a positive number is always its principal root (a positive number) when one exists.
However, most radicals cannot be simplified so easily. Many whole numbers and fractions do not have roots that are also whole numbers or fractions. You can simplify the original radical, but you will still have a radical in your answer.
To simplify a single radical, such as √32, find two factors of the radicand, one of which is a perfect square: √32 = (√16)(√2). Notice that √16 is a perfect square; the positive square root of 16 is 4. So, √32 = (√16)(√2) = 4√2.
Rationalizing Denominators of Fractions
An expression is not in simplest form if there is a radical in the denominator of a fraction. For example, the fraction : is not in simplest form. Multiply the top and bottom of the fraction by the radical in the denominator: Multiply by . Because = 1, this will not change the value of the fraction. Because any radical multiplied by itself is equal to the radicand, (√3)(√3) = 3; (4)(√3) = 4√3, so the fraction : in simplest form is :.
Solving Equations with Radicals
Use the properties of adding, subtracting, multiplying, dividing, and simplifying radicals to help you solve equations with radicals. To remove a radical symbol from one side of an equation, you can raise both sides of the equation to a power. Remove a square root symbol from an equation by squaring both sides of the equation. Remove a cube root symbol from an equation by cubing both sides of the equation.
If √x = 6, what is the value of x?
To remove the radical symbol from the left side of the equation, square both sides of the equation. In other words, raise both sides of the equation to the power that is equal to the root of the radical. To remove a square root, or second root, raise both sides of the equation to the second power. To remove a cube root, or third root, raise both sides of the equation to the third power.
√x = 6, (√x)2 = (6)2, x = 36
= 3, ()3 = (3)3, x = 27
When a value, or base, is raised to a power, that power is the exponent of the base. The exponent of the term 42 is 2, and the base of the term is 4. The exponent is equal to the number of times a base is multiplied by itself: 42 = (4)(4); 26 = (2)(2)(2)(2)(2)(2).
Tip: Any value with an exponent of 0 is equal to 1: 10 = 1, 100 = 1, x0 = 1.
Tip: Any value with an exponent of 1 is equal to itself: 11 = 1, 101 = 10, x1 = x.
An exponent can also be a fraction. The numerator of the fraction is the power to which the base is being raised. The denominator of the fraction is the root of the base that must be taken. For example, the square root of a number can be represented as , which means that x must be raised to the first power (x1 = x) and then the second, or square, root must be taken: = √x.
= (√4)3 = 23 = 8
It does not matter if you find the root (represented by the denominator) first, and then raise the result to the power (represented by the numerator), or if you find the power first and then take the root.
= (√4)3 = 64, √64 = 8
A base raised to a negative exponent is equal to the reciprocal of the base raised to the positive value of that exponent.
Multiplying and Dividing Terms with Exponents
To multiply two terms with common bases, multiply the coefficients of the bases and add the exponents of the bases.
(3x2)(7x4) = 21x6
(2x–5)(2x3) = 4x–2, or
(xc)(xd) = xc + d
To divide two terms with common bases, divide the coefficients of the bases and subtract the exponents of the bases.
= x c – d
Raising a Term with an Exponent to Another Exponent
When a term with an exponent is raised to another exponent, keep the base of the term and multiply the exponents.
(x3)3 = x9
(xc)d = xcd
If the term that is being raised to an exponent has a coefficient, be sure to raise the coefficient to the exponent as well.
(3x2)3 = 27x6
(cx3)4 = c4x12
Find practice problems and solutions for these concepts at Polynomials and Radicals Practice Problems.