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ZingPath: Polynomial Operations

Polynomial Long Division

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Polynomial Long Division


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Divide a polynomial P(x) by another polynomial D(x) using long division.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Divide a polynomial P(x) by D(x) using long division.

Everything You'll Have Covered

Dividing polynomials uses the same process as dividing whole numbers.

When we divide two numbers, we have a dividend and a divisor. The dividend is the number that goes under the division bar, and the divisor goes to the left of the division bar:

In long division with whole numbers, we attempt to divide the first digit of the dividend by the divisor. If this is unsuccessful, we try to divide the first two digits of the dividend by the divisor. In the above example, we get to this point by dividing 35 by 16:

When we continue the process, we subtract 35 from 32 (which results in 3), and then bring the last 3 in 353 down. We now divide 33 by 16, and get to this point:

In this example, 22 is the quotient, and 1 is the remainder. We can rewrite our problem as follows: 353 = 16 22 + 1

Now, let's apply this process to the division of polynomials.

We have a dividend and a divisor in our problem. The dividend is placed under the division bar, and the divisor is to the left:

We divide the first term of the dividend by the first term of the divisor:

Then, we multiply the first term of the quotient (which is above the division bar) by all the terms in the divisor. The resulting polynomial is subtracted from the dividend:

The process is repeated until there is nothing left in the dividend to bring down or the degree of the remainder is less than the degree of the divisor.

When the problem is complete, we can write our answer in the form P(x) = Q(x) . D(x) + R(x) . R(x) is written as a fraction, with the remainder as the numerator and the divisor as the denominator:

Law of Exponents

In polynomial long division, the Law of Exponents is .

This Law of Exponents is used in dividing the first term of the dividend by the divisor.

For example:

This can be rewritten according to the Law of Exponents as:

In order to figure out what the missing number is, the Law of Exponents states that we add the exponents:

The following key vocabulary terms will be used throughout this Activity Object:

  • degree - the greatest exponent or combination of exponents of one single term of a polynomial
  • dividend- a polynomial that is to be divided by another polynomial of equal or lesser degree
  • divisor - a polynomial that is divided into another polynomial of equal or greater degree
  • long division - a process of division, usually used when the divisor is a large number or polynomial, in which each step of the division is written out
  • polynomial - a mathematical expression consisting of a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient
  • quotient - the polynomial that results from the division of one polynomial (the dividend) by another polynomial (the divisor) of equal or lesser degree
  • remainder - the number or polynomial left over after long division is complete

Tutorial Details

Approximate Time 15 Minutes
Pre-requisite Concepts Basic operations of polynomials (adding, subtracting, and multiplying), Laws of Exponents, whole number division
Course Algebra-2
Type of Tutorial Skills Application
Key Vocabulary dividing polynomials, long division, polynomials