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For the Algebra 2 category

Jul 21

The Most Common Factoring Mistake

By Tutor GuyNo Comments

 

I am surprised (and a little disappointed) every year when one of my students tries to simplify a polynomial fraction by cancelling out terms that can’t be cancelled out. For example, when faced with

\dfrac{3x+7}{2x-5}

inevitably, a student will ask me, “Can I cancel out the x’s like this?”

 

 

Tears well up in my eyes as I explain that no, the x’s do not cancel. I explain patiently why the x’s do not cancel. And very often, the next time I work with that student, he or she will try to cancel out the x’s again. This is the most common factoring mistake I see students make, and it’s not limited to Algebra students. I’ve even seen Calculus student make this error. That usually makes me sob quite loudly.

If you would like to keep me from crying, then you need to learn how to simplify polynomial fractions. It’s quite simple once you understand that terms that are added do not cancel out. Only factors that are multiplied together can cancel. Let’s start by looking at a fraction with numbers and no variables.

\dfrac{210}{462}

 Can this be simplified? Of course. Most students will divide the top and bottom by 2, then by 3, and then by 7, as follows:

\dfrac{210}{462}=\dfrac{105}{231}=\dfrac{35}{77}=\dfrac{5}{11}

 This is correct. But why can you cancel out a 2 and a 3 and a 7? It’s because they are factors of the numerator and the denominator. Let’s do the same problem by completely factoring the top and bottom first:

\dfrac{210}{462}=\dfrac{2\cdot 3 \cdot 5 \cdot 7}{2 \cdot 3 \cdot 7 \cdot 11}=\dfrac{5}{11}

 

When you write it out like this, you can see that the 2’s cancel, the 3’s cancel, and the 7’s cancel. And they cancel only because they are factors.

Now let’s try to simplify some polynomial fractions. Start by factoring the numerator and denominator completely, then any like factors will cancel:

\dfrac{2x-4}{6x+14}=\dfrac{2(x-2)}{2(x+7)}=\dfrac{x-2}{x+7}

\dfrac{4x+8}{12x+24}=\dfrac{4(x+2)}{12(x+2)}=\dfrac{4}{12}=\dfrac{1}{3}

\dfrac{x^2-1}{x^2+4x+3}=\dfrac{(x+1)(x-1)}{(x+1)(x+3)}=\dfrac{x-1}{x+3}

 

Dec 19

Translating functions

By Tutor GuyNo Comments

 

Any function f(x) is translated h units to the right when you replace x with x-h. Any function is translated k units up when you replace y with y-k. This is the basis of the vertex form of parabolas and standard form of the other conic sections, but it will help you graph almost any function you encounter without having to write out a “tree” first. If you know what the “parent” functions look like (and by the time you get to Algebra 2, you should memorize the parent functions), then you can graph translations without too much trouble.

Example 1: Graph

 f(x) = ln (x-3).

Here the blue dotted graph is the parent function f(x) = ln (x). The red graph shows the solution, by translating the graph 3 units to the right.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In the graphs below, the parent function has been translated 3 units to the left and 2 units up.

Oct 24

Factoring the sum or difference of two cubes

By Tutor GuyNo Comments

 

The sum or difference of two cubes can always be factored as follows:

(x^3+y^3)=(x+y)(x^2-xy+y^2)

(x^3-y^3)=(x-y)(x^2+xy+y^2)

 These formulas are not too easy to figure out on your own, so your best approach is to memorize them. But how do you keep straight where the plus and minus signs go? Some students like to use the acronym “SOAP”, which stands for Same-Opposite-Always Positive. This means that the first sign is the same as the sign in the sum or difference of the cubes, the second sign is the opposite of this sign, and the third sign is always positive.

Oct 20

Scientific notation on your calculator

By Tutor GuyNo Comments

 

To express numbers in scientific notation on your calculator, you should always use the EE key (EXP key on some calculators). Some calculators will insert a small capital E, others will display the exponents on the far right side of your calculator display. Remember that the E means “times ten to the…”. You should never write scientific notation numbers on your calculator with the caret key (such as 3.1 x 10^4).

Note that a number like (2.3)4 is not in scientific notation! It is not the same number as 2.3 x 104. The number in red is equal to 2.3 x 2.3 x 2.3 x 2.3. The number in blue is equal to 2.3 x 10,000. So for a number like (2.3)4, do not use the EE key.

Most scientific and graphing calculators let you change the display back and forth between regular notation and scientific notation. If you learn how to use this feature, you don’t have to count zeroes when you get an answer like 0.00000571. Just convert to scientific notation and the calculator tells you the number is 5.71 x 10-6.

Oct 17

Solving sin x equations

By Tutor GuyNo Comments

 

“Sine” is a function, not a number. Sin (x) does not mean ‘sine times x’. New trig students often want to solve an equation like sin x = 0.5 by ‘dividing by sine’. This would be like solving

\sqrt{x}=6 \text{ as } x= \dfrac{6}{\sqrt{\text{ }}}                           

 It’s completely meaningless! Instead, you need to take the inverse sine of both sides:

\sin x=0.5

\sin^{-1} \sin x = \sin^{-1} 0.5=30 \textdegree

Oct 13

What are inverse functions?

By Tutor GuyNo Comments

 

The inverse of a function undoes whatever the original function did. If you plug a number into a function and then plug the result of this into the inverse, you always get back what you started with.

Think of a function as a machine that operates on a number. You put in a value for x, the machine performs the rule defined by f(x), and the machine then spits out the new value.

 

 

 

The inverse function, written f -1(x), is the function that reverses all the operations and gives you the original number back. You can think of it as turning the function machine upside down and running the number through it backwards:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

As an example, let’s say that f(x) = 2x + 4. If you put in 1, the machine gives out 6; if you put in 10, the machine gives out 24; if you put in -5, the machine gives -6.

 

 

 

 

 

 

 

 

 

 

 

 

 

The inverse function for 2x + 4 is ½x – 2. If you put 6, 24, and -6 into this function, you get back the numbers you started with!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

You can always show that two functions are inverses of each other by showing that
f(f ‑1(x)) = x or f ‑1(f(x)) = x. Using the example above:

f(f -1(x)) = 2(½x – 2) + 4 = x – 4 + 4 = x.

Sep 05

Exponents and logarithms

By Tutor GuyNo Comments

 

The two equations y = bx and logb y = x are equivalent. They mean the same thing.

The first equation is called the exponential form and the second is called the logarithmic form. Many logarithm problems can be made much simpler by converting them to exponential form. If you can convert back and forth between these two forms, you will find exponential and logarithm problems easier to solve.

Example 1: Solve: log2 (x2) = 6.

Solution:  Rewrite the problem in exponential form:

26 = x2 → 64 = x2 → x = ±8.

Example 2: Solve: ex = 35.

Solution: Rewrite the problem in logarithmic form:

ln 35 = x  → x ≈ 3.56.

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