WEEK 1
WEEK 2
WEEK 3
WEEK 4
WEEK 5
WEEK 6
EXTRA

Unit 51: Equal Polynomials

There is a rule that can help solve what would otherwise be difficult algebra problems on the SAT.  You might find several places on your test where it’s applicable.  The rule is:

When two polynomial equations are set equal to each other, the (boldsymbol x^{mathbf2}) coefficients MUST be equal to each other.

This makes logical sense, as the (x^2) coefficient controls whether a parabola opens up or down, and its width.  If equations are equal, it means their lines on the graph are exactly the same, so the (x^2) coefficients have to be the same.

Here is an example:

$$8x^2+15x-56=abx^2+10x-20x+20+25x-76$$

In the equation above, if b = 4, what does a equal?

Strategy:

  1. Foil or distribute if one side is factored.
  2. Identify which coefficients on the left side of the equals sign correspond to which coefficients on the right side. For example, (mathbf5boldsymbol x^{mathbf2}boldsymbol=boldsymbol aboldsymbol x^{mathbf2}), the 5 and the a correspond.
  3. Write out mini equations of things that are equal. With (mathbf5boldsymbol x^{mathbf2}boldsymbol=boldsymbol aboldsymbol x^{mathbf2}) you would write (mathbf5boldsymbol=boldsymbol a).
  4. Form systems of equations if necessary.

Now back to our example:

We have quadratics set equal to each other.  So, the (x^2) coefficients must be equal.

$$8x^2=abx^2$$

We also know that b = 4, so substituting gives us:

$$8x^2=4ax^2$$

In order for these to be equal, a must be 2.

Here’s a slightly less obvious usage of the rule.

$$4x^2-9=(px+t)(px-t)$$

In the equation above, p and t are constants.  Which of the following could be the value of p ? 

A) 2 

B) 3 

C) 4 

D) 9 

If we FOIL the right side, we’ll get

$$p^2x^2-t^2$$

By our rule, we know the (x^2) coefficients must be equal.  So:

$$p^2=4$$

Taking the square root of both sides tells us p could be 2.