GED Mathematical Reasoning: Simplifying Polynomials

Example 1

Simplify: 3x^2 - 10xy - 5 - 4y^2 + 13xy + 12y^2 - 2x^2

 

The polynomial shown here in example one contains 7 terms.

Remember, the sign in front belongs to the term. If there is an addition sign in front, the term is positive. If there is a subtraction sign, the term is negative.

The individual terms in the example are:  3x^2, - 10xy, - 5, - 4y^2,  13xy, 12y^2, - 2x^2

To simplify the polynomial, we will identify the “like terms” and combine them. The like terms in this example are: 

3x^2 and -2x^2
-10xy and 13xy
-4y^2 and 12y^2
-5

When simplifying a polynomial, you may find it helpful to underline “like terms” in a unique way, like I’ve done here. And remember:

algeb26

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

-10xy + 13xy = (-10 + 13)xy = +3xy
-4y^2 + 12y^2 = (-4 + 12)y^2 = +8y^2
-5 \leftarrow include “as is” in our final answer

As a suggestion, combine “like terms” separately on your paper as I have shown here. This is sometimes referred to as the “stack” method because we’ve ‘stacked’ the combinations vertically – one below the other.

Including the sign of each combination may help, too, since – when you put everything together for your final answer – positive terms will be preceded by an addition sign and negative terms will be preceded by a subtraction sign. The only exception to this is the first leading term. If the leading term is negative, it will be preceded by a negative sign. If it is positive, there will be no sign in front.

We’re finally ready to state the final answer for our example.

The original polynomial in our first example was: 3x^2 - 10xy - 5 - 4y^2 + 13xy + 12y^2 - 2x^2

After combining “like terms,” it simplifies to: x^2 + 3xy + 8y^2 - 5

Notice that positive terms are preceded by an addition sign and negative terms are preceded by a subtraction sign. The only exception to this is the first leading term. Since it is positive, there is no sign in front.

 

Example 2

Simplify: 4a^2 + 6b^2 - 2a

 

The polynomial in Example 2 contains three terms: 4a^2, 6b^2, -2a Do you see that none of these terms are “like terms”? Each term contains a different combination of variables and exponents – none of the variable parts match.

Although the terms:   4 “a” squared and -2 “a” both contain the variable “a”, one contains an exponent of 2 and the other does not. Since none of the terms are “like terms” this means none of the terms can be combined and the polynomial cannot be simplified further.

4a^2 + 6b^2 - 2a \leftarrow cannot be simplified

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