Algebra I All-in-One For Dummies. Mary Jane Sterling
is an expression containing variables and constants. It consists of one or more terms. The terms are separated by addition and subtraction. And the exponents on the variable terms are always whole numbers, never fractions or negative numbers.Ex:
A monomial is a polynomial consisting of exactly 1 term.Ex:
A binomial is a polynomial consisting of exactly 2 terms.Ex:
A trinomial is a polynomial consisting of exactly 3 terms.Ex:
A linear expression is a polynomial in which there is no variable with an exponent greater than 1. In fact, the exponents can be only 1 or 0. And there must be at least one variable term with an exponent of 1.Ex:
A quadratic expression is a polynomial in which there is no variable with an exponent greater than 2. In fact, the exponents can be only 2, 1, or 0. And there must be at least one variable term with an exponent of 2.Ex.
Relating operations with symbols
The basics of algebra involve symbols. Algebra uses symbols for quantities, operations, relations, or grouping. The symbols are shorthand and are much more efficient than writing out the words or meanings. But you need to know what the symbols represent, and the following list shares some of that information. The operations are covered thoroughly in Chapter 6.
+ means add or find the sum or more than or increased by; the result of addition is the sum. It also is used to indicate a positive number.
– means subtract or minus or decreased by or less; the result is the difference. It’s also used to indicate a negative number.
× means multiply or times. The values being multiplied together are the multipliers or factors; the result is the product. Some other symbols meaning multiply can be grouping symbols: ( ), [ ], { }, ·, *. In algebra, the × symbol is used infrequently because it can be confused with the variable x. The symbol is popular because it’s easy to write. The grouping symbols are used when you need to contain many terms or a messy expression. By themselves, the grouping symbols don’t mean to multiply, but if you put a value in front of or behind a grouping symbol, it means to multiply.
÷ means divide. The number that’s going into the dividend is the divisor. The result is the quotient. Other signs that indicate division are the fraction line and slash, /.
means to take the square root of something — to find the number, which, multiplied by itself, gives you the number under the sign. (See Chapter 6 for more on square roots.)
means to find the absolute value of a number, which is the number itself or its distance from 0 on the number line. (For more on absolute value, turn to Chapter 2.)
π is the Greek letter pi that refers to the irrational number: 3.14159. It represents the relationship between the diameter and circumference of a circle.
is the greatest integer operation. It tells you to evaluate what’s in the brackets and replace it with the biggest integer that is not larger than what’s in them.
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Q. Use mathematical symbols to write the expression: “The absolute value of the sum of x and 8 times the greatest integer value of the quotient of x and 3.”
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14 Use mathematical symbols to write the expression: “The difference between x and two is divided by pi.”
15 Use mathematical symbols to write the expression: “The product of six and the absolute value of the sum of eight and y is divided by the square root of the difference between 9 and x.”
Taking Aim at Algebra-Speak
Everything you study requires some understanding of the vocabulary and any special notation. When you can use one word like “introductory” instead of “all that good stuff that comes before the meat of the matter,” then you’ve saved time and space and gotten to the point quickly.
Algebra is full of good words and symbols, as you see in the previous section. And now you find how specific symbols and wording gets right to the point (and, yes, a point in algebra can mean multiply). You’re “equal” to the challenge!
Herding numbers with grouping symbols
Before a car manufacturer puts together a car, several different things have to be done first. The engine experts have to construct the engine with all its parts. The body of the car then has to be mounted onto the chassis and also secured. Other car assemblers have to perform the tasks that they specialize in as well. When these tasks are all accomplished in order, the car can be put together. The same is true with algebra. You have to do what’s inside the grouping symbol before you can use the result in the rest of the equation.
Grouping symbols tell you that you have to deal with the terms inside the grouping symbols before you deal with the larger problem. If the problem contains grouped items, do what’s inside a grouping symbol first, and then follow the order of operations. The grouping symbols are as follows.
Parentheses ( ): Parentheses are the most commonly used symbols for grouping.
Brackets [ ] and braces { }: Brackets and braces are also used frequently for grouping and have the same effect as parentheses. Using the different types of symbols helps when there’s more than one grouping in a problem. It’s easier to tell where a group starts and ends.
Radical : This is used for finding roots.
Fraction line (called the vinculum): The fraction line also acts as a grouping symbol — everything above the line (in the numerator) is grouped together, and everything below the line (in the denominator) is grouped together.
Even though the order of operations and grouping-symbol rules are fairly straightforward, it’s hard to describe, in words, all the situations that can come up in these problems. The explanations and examples in Chapters 3 and 7