145 LearnersLast updated on 5 August 2025
Subtracting Polynomials
A method where the polynomials are subtracted by converting signs to opposites is known as subtracting polynomials. Depending on the expressions, it is similar to addition of polynomials. The positive signs must be changed to negatives and vice versa.
What is Subtracting Polynomials?
Subtraction of polynomials is done using two methods, vertical and horizontal. For simplification, like terms in a polynomial are separated and aligned together. Columns help in matching correct terms during complicated subtractions and this is especially done in the vertical method.
Two rules while subtracting polynomials are:
- Like terms should always be together.
- The signs of all terms being subtracted must change, i.e., all terms having negative signs changes to positive and vice versa.
What are the Steps to Subtract Polynomials?
Polinomial dikurangkan menggunakan metode vertikal atau metode horizontal. Dalam metode horizontal pengurangan polinomial, tanda-tanda suku yang berada dalam tanda kurung pada ekspresi kedua berubah. Ini menyederhanakan pengurangan sehingga dapat diselesaikan sebagai penjumlahan. Dalam metode vertikal, polinomial disusun dalam kolom satu di atas yang lain berdasarkan suku sejenis. Tanda-tanda kemudian diubah sesuai, dan pengurangan dilanjutkan.
Mari kita pahami langkah-langkah yang digunakan dalam kedua metode:
Metode 1: Metode horizontal
- Pastikan polinomial dalam bentuk standar.
- Tempatkan polinomial bersebelahan.
- Ubah tanda untuk semua suku dalam tanda kurung. Lakukan ini untuk polinomial kedua.
- Pisahkan dan susun suku sejenis bersama-sama.
- Hitung suku-suku untuk menentukan hasilnya.
Mari kita terapkan langkah-langkah ini pada contoh:
Soal: Kurangkan 2a + 7b -3c dari 6a - 4b + 5c
Karena polinomial sudah dalam bentuk standar, mari kita mulai dengan langkah 2, yaitu menempatkannya secara horizontal.
(6a - 4b + 5c ) - ( 2a + 7b - 3c )
Langkah 3: Ubah tanda
6a - 4b + 5c - 2a - 7b + 3c
Langkah 4: Susun suku sejenis bersama-sama
6a - 2a - 4b - 7b + 5c + 3c
Langkah 5: Selesaikan ekspresi
4a - 11b + 8c
Oleh karena itu, setelah mengurangkan 2a + 7b -3c dari 6a - 4b + 5c, kita mendapat 4a - 11b + 8c.
Metode 2: Metode vertikal
- Susun polinomial dalam bentuk standar.
- Tempatkan secara vertikal. Suku sejenis ditempatkan satu di atas yang lain.
- Dalam kasus suku yang hilang, gunakan 0 sebagai koefisien untuk mempertahankan keselarasan dalam persamaan.
- Ubah tanda suku-suku dalam polinomial kedua.
- Hitung
Misalnya: Kurangkan 6 + 3x2 - 5x dari -2x + 4 - x2
Langkah 1: Susun polinomial dalam bentuk standar
Polinomial pertama : -2x + 4 - x² → -x² - 2x + 4
Polinomial kedua: 6 + 3x2 - 5x → 3x² - 5x + 6
Langkah 2: Susun suku sejenis secara vertikal
-x2 - 2x + 4
-(3x2 - 5x + 6)
Langkah 3: Karena kedua polinomial memiliki suku untuk x2, kita tidak perlu menggunakan 0 sebagai koefisien dalam kasus ini.
Langkah 4: Ubah tanda
-x2 - 2x + 4
-3x2 + 5x - 6
Langkah 5: Hitung
(-x² - 3x²) = -4x²
(-2x + 5x) = 3x
(4 - 6) = -2
Oleh karena itu, mengurangkan polinomial yang diberikan secara vertikal, kita mendapat jawaban: -4x2 + 3x - 2
Real-life Applications of Subtracting Polynomials
We subtract polynomials to simplify and solve expressions. Subtracting polynomials can also be used to solve real-life situations like:
Calculating usable floor area in architecture
Polynomial subtraction is used to calculate areas. Architects find the usable floor area of staircases, columns, etc., to plan a building's layout accordingly.
Calculating net profit in a business
Business models like revenue and cost have changing values that are modelled as polynomials. The cost model can be subtracted from revenue to find net profit. This process requires polynomial subtraction.
Relative velocity in physics
In motion-related problems, relative velocity is found using polynomial subtraction of velocities of two objects.
Calculating drug concentration in a patient’s body
Pharmacologists use polynomial subtraction to find portions of drugs that have been metabolized in a patient's body over time. This is required for safe drug dosages at regular intervals.
Stock tracking in inventory management
The number of items in stock and sold is expressed as polynomials. Retailers use polynomial subtraction to determine stock levels and make restocking decisions.
Common Mistakes and How to Avoid Them in Subtracting Polynomials
Performing algebraic operations with polynomials can be a little confusing in the beginning. Here are a few common errors related to subtraction of polynomials and how they can be avoided.
Mistake 1
Not changing signs of the second polynomial
Do not forget that the sign of each term of the second polynomial must be changed.
Mistake 2
Combining unlike terms
Combine like terms as unlike terms (x2 and x) cannot be combined.
Mistake 3
Errors in rearranging
Pay close attention while writing polynomials in their standard forms and while changing the signs.
Mistake 4
Dropping negative signs
Make sure you don't miss any negative signs while performing calculations.
Mistake 5
Miscalculating arithmetic with coefficients
Do not rush with calculations, check for all signs and coefficients.
Solved Examples of Subtracting Polynomials
Problem 1
Subtract (7x + 4) - (3x - 2)
4x + 6
Explanation
First, we distribute the negative sign and remove the brackets.
7x + 4 − 3x + 2
Then, we combine the like terms
= (7x − 3x) + (4 + 2)
So, (7x + 4) - (3x - 2) = 4x + 6
Problem 2
Subtract (6 + 3x² − 5x) from (−2x + 4 − x²)
−4x2 + 3x − 2
Explanation
Step 1: Arrange the polynomials in their standard form
−x2 − 2x + 4
3x2 − 5x + 6
Step 2: subtract
(−x2 − 2x + 4) − (3x2 − 5x + 6)
Change signs: −x2 −2x + 4 − 3x2 + 5x − 6
Combine like terms: (−x2 − 3x2) + (−2x + 5x) + (4 − 6) = −4x2 + 3x − 2
Problem 3
Subtract (5x^3 + 2x²− 4x + 6) − (3x^3 − x² + x−1)
2x3 + 3x2 − 5x + 7
Explanation
Distribute the minus sign and remove brackets:
(5x3 + 2x2 − 4x + 6) − (3x3 + x2 − x + 1)
Group like terms:
(5x3 -3x3) + (2x2 + x2) + (-4 - x) + (6 + 1)
Simplify all terms:
2x3 + 3x2 - 5x + 7
Problem 4
Subtract (8x^3 + 2 − x) − (5x^3 + 4x)
3x3 − 5x + 2
Explanation
Since neither polynomial has an x2 term, use 0 as the coefficient for the missing term i.e., x2:
(8x3 − x + 0x2 + 2) − (5x3 + 4x + 0x2 + 0)
Distribute minus: 8x3 − x + 2 − 5x3 − 4x
Combine the terms: (8x3 − 5x3) + (−x − 4x) + 2 = 3x3 − 5x + 2
Problem 5
A company's revenue and cost polynomials are: Revenue: R(x) = 4x² + 10x + 100 Cost: C(x) = 3x² + 5x + 60 Find the profit polynomial P(x) = R(x) - C(x)
P(x) = x2 + 5x + 40
Explanation
P(x)=(4x2 + 10x + 100) − (3x2 + 5x + 60)
Change signs: 4x2 + 10x + 100 − 3x2 − 5x − 60
Combine like terms: x2 + 5x + 40
FAQs on Subtracting Polynomials
1.What property is used when subtracting polynomials?
2.How do you simplify the process of subtracting polynomials?
3.Can we subtract polynomials with different degrees?
4.What happens when a variable or exponent is missing?
5.What is the difference between the horizontal and vertical method?
6.How does learning Algebra help students in Indonesia make better decisions in daily life?
7.How can cultural or local activities in Indonesia support learning Algebra topics such as Subtracting Polynomials?
8.How do technology and digital tools in Indonesia support learning Algebra and Subtracting Polynomials?
9.Does learning Algebra support future career opportunities for students in Indonesia?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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