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How To Solve Negative Polynomials

By Kathleen Cantor, 07 Oct 2020

A polynomial is a mathematical expression made of variables and coefficients. The only operations polynomials use are addition, subtraction, positive integer exponents, and multiplication. You cannot raise the variables in a polynomial to irrational powers, complex powers, square roots, etc.

What does a polynomial look like? A simple example:

6x7 + 23x3 – 7

This example is a single-variable polynomial because the only variable in the expression is x.

Even though polynomials exclude negative terms, there are instances when you will encounter negative polynomials. What do you do then? Let's find out.

Important Polynomial Terms

If you want to understand how to solve negative polynomials, you should first know some of the terminology associated with polynomials.

  • Terms of a polynomial. The separate coefficients, variables, and constants that you put together to make a polynomial expression. In the example given earlier, the terms of the polynomial include 6x7, 23x3, and -7. The polynomial has three terms.
  • Variable. This refers to the symbols that are used as placeholders for numbers. In the example above, the variable is x.
  • Coefficient. This is the number that accompanies a variable. In the term 6x7, the coefficient of x7 is 6.
  • Constant. These are numbers in a polynomial that are not associated with a variable. The constant term in the example is -7.
  • Degree of a polynomial. This is the highest power of a variable (the exponent) in the polynomial. The degree of the polynomial is 7.

Some other rules to remember: If a variable does not have a coefficient, then the coefficient is 1. Also, if a variable does not have a power, then the power is also 1.

Negative Polynomials

When you encounter a negative polynomial, you ultimately end up simplifying the expression.

An example of a negative polynomial is

2n-2 – 3n – 7

Notice that the power of n is negative, -2.

To solve or simplify negative polynomials, we have to understand the rules of exponents fully.

The Rules of Exponents

The rules of exponents generalize how to manipulate exponents. Let us review them.

The Zero Exponent

a0 = 1

This means that any number raised to an exponent of 0 is 1.

The Product or Multiplication Rule

ab * ac = ab + c

This means that if you multiply two or more exponents with the same base, the result is simply one exponent (the base) raised to the sum of the powers. Basically, you add the exponents together.

Quotient or Division Rule

ab / ac = ab – c

When you divide two or more exponents with the same base, you subtract the powers.

Negative Exponent Rule

a-b = 1 / ab

When you encounter a number/variable with a negative exponent, take the inverse of the base. In the above example, the inverse of a is 1 / a.

a-b = ( 1 / a )b = 1b / ab = 1 / ab

The Power Rule

( ab )c = ab * c

If you have an exponent that you need to raise to another power, find the product of the two powers.

These rules are important as they will help us simplify negative polynomials. Let's get into some examples to help us better apply these rules.

Example 1

Simplify the expression

2x4 y-2

The expression above is relatively simple. The only thing you need to simplify is the negative exponent. Remember the negative rule.

2x4y-2 = 2x4 / y2

We took the inverse of y since the power is negative.

Example 2

Simplify the expression

 ( 2x−5y−3 * 3x3y−2 ) / ( 3x−5y3 )

This is a bit complicated.

Step 1

This first step is not cast in stone and can be changed depending on the problem. In this case, the numerator contains multiple exponents with the same base, so it is good to start by simplifying the numerator by applying the rules of exponents.

( 2x−5y−3 * 3x3y−2 ) / ( 3x−5y3 )

= ( 2x-5 + 3 * 3y-2 +( -3 ) ) / ( 3x−5y3 )

= ( 6x-2y-5 ) / ( 3x−5y3 )

We have now simplified the numerator. The denominator looks good because the variables there did not occur more than once.

Step 2

Next, simplify the numerator and denominator using the quotient rule.

 ( 6x-2y-5 ) / ( 3x−5y3 )

= ( 6 / 3 ) * x-2 – ( -5 ) * y( -5 ) – 3

= 2x3 * y-8

= 2x3y-8

Step 3

We have eliminated the fraction, however, we still have a negative power. Let us apply the negative exponent rule.


= 2x3 / y8

We have taken the inverse of y. 2x3 is not affected because it does not share the negative power with y.

Example 3

Simplify the expression below.

( 3y-7x3z5 ) / ( 4y-5x2z3 )-2

Step 1

The expression above contains exponents raised to another power, so we have to apply the power rule.

( 3y-7x3z5 ) / ( 4y-5x2z3 )-2

= ( 3y-7x3z5 ) / ( 4-2y10x-4z-6 )

We have multiplied the powers of all the denominators by -2. Remember the power rule.

Step 2

Apply the quotient rule since the variables only appear once in both the numerator and the denominator.

( 3y-7x3z5 ) / ( 4-2y10x-4z-6 )

( 3 / 4-2 ) * ( y-7 – 10 ) * ( x3 – ( -4 )) * ( z5 – ( -6 ))

= ( 3 / 4-2 ) * ( y-17 ) * ( x7 ) * ( z11 )

Step 3

The expression we have arrived at in step 2 has negative exponents.

Apply the negative exponent rule to eliminate them.

( 3 / 4-2 ) * ( y-17 ) * ( x7 ) * ( z11 )

( 3 * 42 ) * ( 1 / y17 ) * x7z11

( 48x7z11 ) / y17

Our final solution is 48x7z11 / y17

Solving, or simplifying, negative polynomials can be complicated. This is why it's so important to understand the different rules of exponents fully. The most important thing, however, when handling negative polynomials is to invert the base whenever you have a negative polynomial. Like any skill, the more you practice it, the better you get. If at first, you do not understand a negative polynomial expression, take your time and check your work -- you'll be a pro in no time!

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