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Subtracting Polynomials
Subtracting polynomials is a method of subtracting polynomials by converting the signs to the opposite sign. The method is similar to adding polynomials. While subtracting polynomials, we convert the sign to either positive or negative depending on the actual equation. Let us learn more about subtracting polynomials, the methods, the steps, and solve a few examples.
1.  What is Subtracting Polynomials? 
2.  Steps to Subtracting Polynomials 
3.  FAQs on Subtracting Polynomials 
What is Subtracting Polynomials?
Subtracting polynomials is very similar to adding polynomials where the minus sign is converted through the parentheses. There are two methods of subtracting polynomials  vertically and horizontally. The like terms in the polynomials are separated to make the subtraction simple. Using columns would help us to match the correct terms together in a complicated subtraction. Keep two rules in mind while performing the subtraction of polynomials:
 Rule 1: Always take like terms together while performing subtraction.
 Rule 2: Signs of all the terms of the subtracting polynomial will change, + changes to  and  changes to +.
Steps to Subtracting Polynomials
Subtracting polynomials is done in two methods  vertically and horizontally. In both methods, the rules mentioned above are applicable. In the horizontal method, the signs through the parentheses of the second equation change to perform simple addition. In the vertical method, the equations are written columnwise according to the terms, the signs are converted and subtraction takes place. Let us see the steps to perform subtraction of polynomials in both methods.
Method 1: Subtracting Polynomials Horizontally
 Step 1: Arrange the polynomials in their standard form.
 Step 2: Place the polynomial next to each other horizontally.
 Step 3: Change the signs of the second polynomial through the parentheses to its opposite sign.
 Step 4: Separate the like terms and arrange them together.
 Step 5: Perform the calculations.
Let us look at an example to understand this better.
For example: Subtract 4x  10y + 15z from 5x + 8y  20z
Step 1: Arrange the polynomials in standard form. In this example, it is already arranged.
Step 2: Place them horizontally.
(4x  10y + 15z)  (5x + 8y  20z)
Step 3: Change the sign of the second polynomial through the parentheses.
4x  10y + 15z  5x  8y + 20z
Step 4: Arrange the like terms together.
4x  5x  10y  8y + 15z + 20z
Step 5: Solve the equation
x  18y + 35z
Therefore, when we subtract 4x  10y + 15z from 5x + 8y  20z = x  18y + 35z.
Method 2: Subtracting Polynomials Vertically
Subtracting polynomials vertically is done in a very simple manner. Once the polynomials have been arranged vertically, the signs in the second polynomials are converted from plus to minus and minus to plus without the requirement of the parentheses. This is also a shortcut method for subtracting two or more polynomials. Polynomials can be subtracted in vertical arrangement using the steps given below,
 Step 1: Arrange the polynomials in their standard form
 Step 2: Place the polynomials in a vertical arrangement, with the like terms placed one above the other in both the polynomials.
 Step 3: We can represent the missing power term in the standard form with "0" as the coefficient to avoid confusion while arranging terms.
 Step 4: The signs of the terms in the second polynomials change as done in a horizontal manner.
 Step 5: Perform the calculations
Let us look at an example.
For example: Subtract 4x^{2} + 8x + 10 from 5x^{2} 14x  15
 Step 1: Arranging the polynomial in standard form. In this case, they are already in their standard forms.
 Step 2: Like terms in the above two polynomials are: 4x^{2} and 5x^{2}, 8x and 14x, 10 and 15.
 Step 3: Enclose the part of the polynomial which to be deducted in parentheses with a negative () sign prefixed. Then, remove the parentheses by changing the sign of each term of the polynomial expression.
 Step 4: Calculate after altering the signs of the subtracting polynomials:
Related Topics
Listed below are a few topics that are related to subtracting polynomials, take a look
Examples on Subtracting Polynomials

Example 1: Subtract the polynomials horizontally 16x^{3} + 32y^{2}  8z  7 from 27x^{3}  4y^{2} + 31z + 9
Solution: Arrange the polynomials horizontally.
(27x^{3}  4y^{2} + 31z + 9)  (16x^{3} + 32y^{2}  8z  7)
Change the signs through the parentheses and remove the terms from the parentheses.
27x^{3}  4y^{2} + 31z + 9  16x^{3}  32y^{2} + 8z + 7
Arrange the like terms together.
27x^{3}  16x^{3}  4y^{2}  32y^{2} + 31z + 8z + 9 + 7
11x^{3}  36y^{2} + 39z + 16
Therefore, the answer is 11x^{3}  36y^{2} + 39z + 16.

Example 2: Subtract the polynomials vertically x^{2}  45y + 35 and 7x^{2}  12y + 10.
Solution: Arrange the polynomials vertically according to like terms, change the signs of the second polynomial, and calculate.

Example 3: Subtract the polynomials vertically (5x^{2} – 17x – 41) – (x^{3} – 7x^{2} + 4) + (9x^{3} + 14x^{2} – 12x – 51)
Solution: Arrange the polynomials vertically according to like terms, change the signs of the second polynomial, and calculate. Changing the signs for the second polynomial as there is a negative sign before the second polynomial, hence we get x^{3} + 7x^{2}  4.
FAQs on Subtracting Polynomials
What is Meant by Subtracting Polynomials?
Subtracting polynomials is very similar to adding polynomials but while subtracting we change the signs from plus to minus and minus to plus. Subtracting polynomials is done in two methods  vertically and horizontally. In both methods, when a minus sign is mentioned before the parentheses, we always change the signs.
What is the Rule in Subtracting Polynomials?
While subtracting polynomials, the two important rules are:
 Rule 1: Always take like terms together while performing subtraction.
 Rule 2: Signs of all the terms of the subtracting polynomial will change, + changes to  and  changes to +.
What Property is Used When Subtracting Polynomials?
While subtracting polynomials, we can use the distributive property. This is even applicable while adding polynomials. For example (3x^{2} + 2x + 6) + (x^{2}  6x  10) = (3 + 1)x^{2} + (2  6)x + (6  10) = 4x^{2}  4x  4.
What are the Steps of Subtracting Polynomials Horizontally?
The important step before subtracting a polynomial is converting the subtraction problem to an addition problem by converting minus to plus and plus to minus. The steps to subtract polynomials horizontally are:
 Arrange the polynomials in their standard form.
 Place the polynomial next to each other horizontally.
 Change the signs of the second polynomial through the parentheses to its opposite sign.
 Separate the like terms and arrange them together.
 Perform the calculations.
What are the Steps of Subtracting Polynomials Vertically?
Subtracting polynomials vertically is also done by converting the problem from subtraction to addition. Arrange the polynomials vertically or columnwise one below the other according to the like terms and calculate. Here are the steps:
 Arrange the polynomials in their standard form
 Place the polynomials in a vertical arrangement, with the like terms placed one above the other in both the polynomials.
 We can represent the missing power term in the standard form with 0 as the coefficient to avoid confusion while arranging terms.
 The signs of the terms in the second polynomials change as done in a horizontal manner.
 Perform the calculations.
How Do You Simplify Subtracting Polynomials?
To simplify subtracting polynomials, the first step is to convert the problem from subtraction to addition. The signs in the second polynomial while placed horizontally are converted through the parentheses i.e. minus to plus and plus to minus. Once the polynomials are converted, normal addition of polynomials is performed.
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