MA103 – Intermediate Algebra

Tips – Hints – Pitfalls – Formulas - Things to know, remember, and love!

1. When you multiply or divide an inequality by a negative number, reverse the direction of the inequality. (Flip it!)

2. Recognize and factor the sum or difference of two cubes. Signs matter.

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3. KNOW THE QUADRATIC FORMULA

4. BE CAREFUL!!!! That’s a DOES NOT equal sign!! You are missing the “middle” term.

This can be generalized: . The left side here DOES NOT equal the right side. You are missing several terms now .

Signs Matter!

5. The sum of two squares, does not factor over the set of real numbers.

6. BUT, the difference of two squares does factor:

7. The power rule lets you raise both sides of an equation to the same power.

Do NOT apply this term-by-term.

8. KNOW the Properties of Logarithms

9. With ease and grace, move from the logarithmic form of an equation to the exponential form, and vice versa.

or If , then

10. There are two frequently used logarithmic bases. Your calculator probably has one key for each.

· A log expression written without a base is a common logarithm and the implied base is 10.

It follows from the properties listed in #8 above, that

· The natural log base is e and is written as ln.

It follows from the properties above that

11. Change of base formula. This is useful if you want to use your calculator, but your problem does not involve log or ln (common or natural log base).

12. You won’t find this expression in any math book, but you may “unlog” both sides of an equation.

For example:

Solving the quadratic equation yields:

CAUTION!! The logarithm of a negative number is not a real number.

Check your “solutions”! Neither nor is a real number, so

is not an acceptable solution. Only x = 2 checks if we limit ourselves to the set of real numbers. x = 2 is the only acceptable solution.

One more cautionary note: Just as you may only take the log of the entire side of an equation, you may only “unlog” if each side of an equation contains only a single log expression.

BE CAREFUL. does NOT equal c.

13. Radicals can be expressed with rational exponents.

For example:

14. Negative exponents DO NOT make the number negative.

If you change the location of the expression with the exponent, change the sign of the exponent.

15. To clear the denominators (or eliminate the denominators) from an equation, multiply both sides, ALL TERMS, by the LCD of both sides.

16. To add/subtract rational numbers/expressions, multiply the numerator and denominator of each term by its very own “missing factors.”

Notice that this rational expression could be reduced to .

17. KNOW the Pythagorean Theorem! Know that it applies only to right triangles. Know that a right triangle contains a 90° (right) angle.

a c = hypotenuse

a and b are the lengths of the “legs” of the triangle and

c is the length of the hypotenuse.

The hypotenuse is the side opposite the right angle. b

It is always the longest side.

This formula, like any formula, can be rearranged. For example: .

18. Quadratic equations may be written in the form

Sometimes this is referred to as the “vertex form” because h and k are the x- and y- coordinates of the vertex, respectively. The x-coordinate of the vertex is found by setting the expression in parentheses equal to zero and solving for x. For example:

In the example on the left, 5 is the x-coordinate and 6 is the y-coordinate. The

vertex is at the point (5, 6).

In the example on the right, (-2, 4) is the vertex.

Notice that the x-coordinate of the vertex has the sign opposite of the operation sign in the equation.

What about the y-coordinate? In the examples above, k was added, just as it is in the stated formula. What happens if k is subtracted?

The vertex in this case is (-7, -5).

19. Synthetic division only works if the divisor (the expression by which you are dividing) is of the form (x – c). What in the world does that mean?

The divisor must be a binomial. It has only 2 terms.

The x-term (or variable term) is only raised to the first power. We don’t bother to show it explicitly when the exponent is 1. But there can be no in the expression. Just x.

The second term, c, or –c determines the number you use as the divisor in synthetic division.

Set the binomial expression equal to zero and solve for x.