Algebra: GED Test Prep (page 5)
When you take the GED Mathematics Exam, you will be asked to solve problems using basic algebra. This article will help you master algebraic equations by familiarizing you with polynomials, the FOIL method, factoring, quadratic equations, inequalities, and exponents.
What Is Algebra?
Algebra is an organized system of rules that help solve problems for "unknowns."This organized system of rules is similar to rules for a board game. Like any game, to be successful at algebra, you must learn the appropriate terms of play. As you work through the following section, be sure to pay special attention to any new words you cmay encounter. Once you understand what is being asked of you, it will be much easier to grasp algebraic concepts.
An equation is solved by finding a number that is equal to an unknown variable.
Simple Rules for Working with Equations
- The equal sign separates an equation into two sides.
- Whenever an operation is performed on one side, the same operation must be performed on the other side.
- Your first goal is to get all of the variables on one side and all of the numbers on the other side.
- The final step often will be to divide each side by the coefficient, the number in front of the variable, leaving the variable alone and equal to a number.
5m + 8 = 48
–8 = –8
m = 8
To check an equation, substitute the number equal to the variable in the original equation.
To check the equation you just solved, substitute the number 8 for the variable m.
5m + 8 = 48
5(8) + 8 = 48
40 + 8 = 48
48 = 48
Because this statement is true, you know the answer m = 8 must be correct.
To learn how to work with percentages or proportions, it is first necessary for you to learn how to cross multiply. You can solve an equation that sets one fraction equal to another by cross multiplication. Cross multiplication involves setting the products of opposite pairs of terms equal.
There is one formula that is useful for solving the three types of percentage problems:
When reading a percentage problem, substitute the necessary information into the previous formula based on the following:
- Always write 100 in the denominator of the percentage sign column.
- If given a percentage, write it in the numerator position of the percentage sign column. If you are not given a percentage, the variable should be placed there.
- The denominator of the number column represents the number that is equal to the whole, or 100%. This number always follows the word of in a word problem.
- The numerator of the number column represents the number that is the percent, or the part.
In the formula, the equal sign can be interchanged with the word is.
Finding a percentage of a given number: What number is equal to 40% of 50?
Finding a number when a percentage is given: 40% of what number is 24?
Finding what percentage one number is of another: What percentage of 75 is 15?
A variable is a letter that represents an unknown number. Variables are frequently used in equations, formulas, and in mathematical rules to help you understand how numbers behave.
When a number is placed next to a variable, indicating multiplication, the number is said to be the coefficient of the variable.
8c 8 is the coefficient to the variable c.
6ab 6 is the coefficient to both variables a and b.
If two or more terms have exactly the same variable(s), they are said to be like terms.
7x + 3x = 10x
The process of grouping like terms together performing mathematical operations is called combining like terms.
It is important to combine like terms carefully, making sure that the variables are exactly the same. This is especially important when working with exponents.
7x3y + 8xy3
These are not like terms because x3y is not the same as xy3. In the first term, the x is cubed, and in the second term, the y is cubed. Because the two terms differ in more than just their coefficients, they cannot be combined as like terms. This expression remains in its simplest form as it was originally written.
A polynomial is the sum or difference of two or more unlike terms.
2x + 3y – z
This expression represents the sum of three unlike terms 2x, 3y, and –z.
Three Kinds of Polynominals
- A monomial is a polynomial with one term, as in 2b3.
- A binomial is a polynomial with two unlike terms, as in 5x + 3y.
- A trinomial is a polynomial with three unlike terms, as in y2 + 2z – 6.
Operations with Polynominals
- To add polynomials, be sure to change all subtraction to addition and the sign of the number that was being subtracted to its opposite. Then, simply combine like terms.
(3y3 – 5y + 10) + (y3 + 10y – 9)
Change all subtraction to addition and the sign of the number being subtracted:
3y3 + –5y + 10 + y3 + 10y + –9
Combine like terms:
3y3 + y3 + –5y + 10y + 10 + –9 = 4y3 + 5y + 1
If an entire polynomial is being subtracted, change all of the subtraction to addition within the parentheses and then add the opposite of each term in the polynomial.
(8x – 7y + 9z) – (15x + 10y – 8z)
Change all subtraction within the parentheses first:
(8x + –7y + 9z) – (15x + 10y + –8z)
Then, change the subtraction sign outside of the parentheses to addition and the sign of each term in the polynomial being subtracted:
(8x + –7y + 9z) + (–15x + –10y + 8z)
Note that the sign of the term 8z changes twice because it is being subtracted twice. All that is left to do is combine like terms:
8x + –15x + –7y + –10y + 9z + 8z = –7x + –17y + 17z is your answer.
To multiply monomials, multiply their coefficients and multiply like variables by adding their exponents.
(–5x3y)(2x2y3) = (–5)(2)(x3)(x2)(y)(y3) = –10x5y4
To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and add the products.
6x(10x –5y + 7)
Change subtraction to addition:
6x(10x + –5y + 7)
(6x)(10x) + (6x)(–5y) + (6x)(7) 60x2 + –30xy + 42x
The FOIL method can be used when multiplying binomials. FOIL stands for 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 like terms of the products.
(3x + 1)(7x + 10)
3x and 7x are the first pair of terms.
3x and 10 are the outermost pair of terms.
1 and 7x are the innermost pair of terms.
1 and 10 are the last pair of terms.
Therefore, (3x)(7x) + (3x)(10) + (1)(7x) + (1)(10) = 21x2 + 30x + 7x + 10.
After we combine like terms, we are left with the answer: 21x2 + 37x + 10.
Factoring is the reverse of multiplication:
Multiplication: 2(x + y) = 2x + 2y
Factoring: 2x + 2y = 2(x + y)
Three Basic Types of Factoring
- Factoring out a common monomial.
- Factoring a quadratic trinomial using the reverse of FOIL:
- Factoring the difference between two perfect squares using the rule:
10x2 – 5x = 5x(2x – 1) and xy – zy = y(x – z)
y2 – y –12 = (y –4)(y + 3) and z2 –2z + 1 = (z –1)(z –1) = (z –1)2
a2 – b2 = (a + b)(a – b) and x2 – 25 = (x + 5)(x – 5)
Removing a Common Factor
If a polynomial contains terms that have common factors, the polynomial can be factored by using the reverse of the distributive law.
In the binomial 49x3 + 21x, 7x is the greatest common factor of both terms.
Therefore, you can divide 49x3 + 21x by 7x to get the other factor.
Thus, factoring 49x3 + 21x results in 7x(7x2 + 3).
A quadratic equation is an equation in which the greatest exponent of the variable is 2, as in x2 + 2x – 15 = 0.A quadratic equation has two roots, which can be found by breaking down the quadratic equation into two simple equations.
Solve x2 + 5x + 2x + 10 = 0.
Combine like terms: x2 + 7x + 10 = 0
Factor: (x + 5)(x + 2) = 0
x + 5 = 0 or x + 2 = 0
Now check the answers.
–5 + 5 = 0 and –2 + 2 = 0
Therefore, x is equal to both –5 and –2.
Linear inequalities are solved in much the same way as simple equations. The most important difference is that when an inequality is multiplied or divided by a negative number, the inequality symbol changes direction.
10 > 5, but if you multiply by –3, (10)(–3) < (5)(–3) –30 < –15
Solving Linear Inequalities
To solve a linear inequality, isolate the letter and solve the same as you would in an equation. Remember to reverse the direction of the inequality sign if you divide or multiply both sides of the equation by a negative number.
If 7 – 2x > 21, find x.
Isolate the variable.
7 – 2x > 21
7 – 2x – 7 > 21 – 7
–2x > 14
Because you are dividing by a negative number, the direction of the inequality symbol changes direction.
x < –7
The answer consists of all real numbers less than –7.
An exponent tells you how many times the number, called the base, is a factor in the product.
Sometimes you will see an exponent with a variable:
The b represents a number that will be a factor to itself n times.
bn where b = 5 and n = 3
Don't let the variables fool you. Most expressions are very easy once you substitute in numbers.
- bn = 53 = 5 × 5 × 5 = 125
Laws of Exponents
- Any base to the zero power is always 1.
- When you multiply identical bases, add the exponents.
- When you divide identical bases, subtract the exponents.
- Here is another method of illustrating multiplication and division of exponents:
- If an exponent appears outside of parentheses, you multiply the exponents together.
50 = 1 700 = 1 29,8740 = 1
22 × 24 × 26 = 212 a2 × a3 × a5 = a10
bm × bn = bm + n
bmbn = bm – n
(33)7 = 321
(g4)3 = g12
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