How to Solve Basic Operations With Polynomials
Math problems and concepts may seem daunting, scary and even sometimes impossible. This is not the case, however. All that is required to master the concepts is learning the steps to solve a concept, and working and reworking problems till the process becomes second nature. Solving polynomials is no exception to this process.
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Instructions
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Adding Polynomials
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1
When adding polynomials, there are two rules to remember. The first is that you must remove the parentheses by simplifying the expression. The second is that you can only combine polynomials that have alike terms. This means that they have the same variables (or letters).
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2
Completing this example, found on brainmass.com, will help you understand the process and the rules: (x^3+4x^2+2x-3) + (-x^3-3x^2-3x+2)
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3
Simplify by multiplying what is inside the parentheses by what is outside. Let's say there was a 2 on the outside of the parentheses, you would multiply all of the coefficients inside by 2. It would look like this: 2 (x^3+4x^2+2x-3). The simplified version would be 2 x^3 + 8 x^2 + 4x - 6. If there is nothing on the outside, like the example given in Step 2, than you can skip this rule. Once they are simplified, remove the parentheses.
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4
Add together the like terms. This means that the letters and exponents are the same.
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5
Look at the example, x^3 + 4x^2 + 2x - 3 - x^3 - 3x^2 - 3x + 2. You can see that there are two x^3 terms: x^3 and -x^3. Remember when adding terms, you are just adding the coefficients. The letters and exponents stay the same.
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6
Add 1 and -1. If there are no visible coefficients in front of a letter, there is an understood 1 there. Adding them together, you get zero, so the x^3 term disappears. We would add the other alike terms in this manner. When you finish adding the like terms, the solution will be: x^2 - x -1
Subtracting Polynomials
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7
The rules for subtracting polynomials are similar to those of adding polynomials. Remove the parentheses by simplifying the expression. Next, combine like terms.
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8
This example, found on brainmass.com, will help to illustrate the rules: (2 - 3x + x^3) - (-1 - 4x + x^2)
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9
Multiply everything inside the parentheses by the number outside the parentheses. This was explained in Step 3 of Adding Polynomials. If there are no numbers outside the parentheses, skip this step.
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10
Reverse the sign of anything inside the parentheses when there is a subtraction sign on the outside. You find this in the example: (2 - 3x + x^3) - (-1 - 4x + x^2). You change all the signs because you are multiplying everything by the negative sign. In the second set of parentheses, the -1 would become a 1, the -4x would become a 4x and the + x^2 would become a - x^2. The simplified expression would be 2 - 3x + x^3 + 1 + 4x - x^2.
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11
Combine the like terms as in Step 5 of Adding Polynomials. The solution would be x^3 - x^2 +x +3
Multiplying Polynomials
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12
Follow this one rule. Multiply all of the alike terms. This includes numbers and letters.
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13
Use this example to help you: -2ab*7a^5*b^4.
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14
Notice there are three terms: -2ab, 7a^5, and b^4. Follow the rule by first multiplying the coefficients together. -2 * 7 is equal to -14. There are no more coefficients, so -14 is your new coefficient for your solution.
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15
Multiply your like letters together. There are tow "a's" in your problem. "a" from the first term, and "a^5" from the second term. According to mathsisfun.com, to multiply two similar letters with different exponents, as is the case, you keep the letter the same, and add the exponents. So our answer would look like a ^ 1+5, or a^6. Multiply the "b's" together to get b^5.
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16
The simplified solution would be: -14 a^6 b^5
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Tips & Warnings
Work through as many examples and problems as you want to until you feel comfortable doing the problems.
References
Resources
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