Transforming Equations (advanced)

Some of these equations can start to get a little more complicated, but remember that they are in fact only testing your isolation skills.

Manipulating Polynomials

The last type of “basic” algebra question you will see requires you to know how to manipulate polynomials.  These are the same rules as the single-value equations, but generally require you to deal with exponents in some manner.

A common question:

Which expression is equivalent to $$left(3x^3-5right)-3(x^3-x^2-2)$$

A) (6x^3+3x^2-7)
B) (-x^2-7)
C) (3x^2+1)
D) (3x^3-3)

Strategy:

  1. Perform your order of operations correctly.  
  2. Combine like terms (same variable and exponent)
  3. Be very careful distributing negatives!

In our example above, they are testing three skills – do you know your order of operations, do you know how to combine like terms, and do you know how to distribute both multiplication AND a +/- sign.

Here, we will first need to distribute our -3. Then, we will need to combine like terms (addition and subtraction fall last). Remember that we can only combine terms of the same variable raised to the same exponent!

Distributing:

$$3x^3-5-3x^3-3x^2+6$$

Combining like terms:

$$3x^2 +1$$

Our answer is C).

Another example the SAT regards as very difficult:

Which of the following is equivalent to (left(a+frac b4right)^2)?

A) (a^2+frac{b^2}4)

B) (a^2+frac{b^2}{16})

C) (a^2+frac{ab}4+frac{b^2}4)

D) (a^2+frac{ab}2+frac{b^2}{16})

When presented with something like this, DO NOT just distribute the exponent!  Notice the answer choices are tempting you to do just that.  Re-write the problem like:

$$left(a+frac b4right)left(a+frac b4right)$$

and then FOIL to get:

$$a^2+frac{ab}4+frac{ab}4+frac{b^2}{16}$$

we then combine our like terms, reduce the middle fraction, and are left with answer D).

Traps

Students miss these problems all the time due to the reasons listed above:  not checking for extraneous solutions, not applying exponent rules correctly, or not keeping close enough track of +/- signs.  If you follow all the algebra rules, these problems are very straightforward.