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# Algebra: GED Test Prep (page 3)

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Updated on Mar 9, 2011

### FOIL

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.

Example

(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

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.
• 10x2 – 5x = 5x(2x – 1) and xyzy = y(xz)

• Factoring a quadratic trinomial using the reverse of FOIL:
• y2y –12 = (y –4)(y + 3) and z2 –2z + 1 = (z –1)(z –1) = (z –1)2

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

Example

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.

Example

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

–5 + 5 = 0 and –2 + 2 = 0

Therefore, x is equal to both –5 and –2.

### Inequalities

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.

Example

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.

Example

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.

### Exponents

An exponent tells you how many times the number, called the base, is a factor in the product.

Example

Sometimes you will see an exponent with a variable:

bn

The b represents a number that will be a factor to itself n times.

Example

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.
• 50 = 1   700 = 1   29,8740 = 1

• When you multiply identical bases, add the exponents.
• 22 × 24 × 26 = 212   a2 × a3 × a5 = a10

• When you divide identical bases, subtract the exponents.
• Here is another method of illustrating multiplication and division of exponents:
• bm × bn = bm + n

bmbn = bmn

• If an exponent appears outside of parentheses, you multiply the exponents together.
• (33)7 = 321

(g4)3 = g12