Adding Polynomials

Adding polynomials is defined as combining like terms, which share the same variable to the same exponent, by adding their coefficients. This combines multiple terms into a single, simplified polynomial. When adding polynomials, there are specific rules to follow that make the operation very easy.

• Rule 1: Like terms are always combined and added. Unlike terms can never be added.

• Rule 2: When adding, keep each term’s sign as is—don’t change it.

Like terms are known as algebraic terms, which have the same variables along with the same exponents. For example, 2x, 7x, and -2x are all like terms.

Unlike terms have different variables or exponents, example 2x, 7x, and –2x. Unlike terms diverge in variable or exponent, those are  2, 7x², and –2y².

There are different methods for adding polynomials. Here are two methods,

• Horizontal
   
• Vertical

to add polynomials along with their steps and example

In the horizontal method, we write all polynomials in one line, separated by signs. Then group like terms (terms with same variables and exponents), and add their coefficients to simplify the expression. 

• Step 1: Horizontally write the polynomials with an addition (+) sign between them.

• Step 2: Combine the like terms by grouping them in parentheses by keeping the sign of each term.

      (4x² −3x + 5) + (2x² + x −7)

• Step 3: Perform the calculations.

 Add the polynomials 4x²+5x−3 and 2x² - 3x +6

Solution:

Step 1: Write the polynomials in horizontal form:

4x² + 5x −3 + 2x² −3x + 6

Step 2: Group like terms together:

(4x² + 2x²) + (5x −3x) + (−3 + 6)

Step 3: Perform the calculations:

6x² + 2x +3

Final Answer: 6x² + 2x + 3

We place the polynomials column-wise vertically. 

Step 1: Write the polynomials in standard form.

Step 2: Place the polynomials in a vertical arrangement, with the like terms.

Step 3: If any power term is missing in any polynomial, we write a '0' as its quantity to avoid confusion in the column-wise arrangement.

Step 4: Perform the calculations by keeping the sign of the terms.

Add the polynomials 4x² + 2x + 1 and 3x² - x + 5

Solution:

Step 1: Write the polynomials in standard form and arrange them vertically:

4x2+2x+1

+3x2-x+5

Step 2: Add the like terms in each column:

• 4x²+3x²=7x²
   
• 2x−x = x
• 1+5 = 6

 4x²   + 3x²

   2x    –  x

    1    + 5

7x²    + x    + 6

Step 3: Write the sum of the polynomials:

7x²+x+6

Final Answer: 7x² + x + 6

Adding polynomials merges related quantities like lengths, prices, or areas by grouping and adding similar terms. This makes real-life tasks more efficient.

• Calculating Total Cost While Shopping: When you buy multiple items, you can add their costs together using a polynomial. For example, if you buy 3 pens at ₹10 each, 2 notebooks at ₹20 each, and 1 eraser at ₹5, the total cost is:Total Cost = 3(10) + 2(20) + 1(5) = ₹30 + ₹40 + ₹5 = ₹75.
• Planning a Birthday Party: If you're organizing a party and need to buy decorations, snacks, and party favors, you can use a polynomial to calculate the total cost. For instance, if decorations cost ₹200, snacks ₹150, and party favors ₹100, the total cost is:Total Cost = 200 + 150 + 100 = ₹450.
• Calculating Total Distance Traveled: If you're going on a trip and want to calculate the total distance, you can add the distances of each leg of the journey. For example, if you travel 50 km, then 30 km, and then 20 km, the total distance is: Total distance = 50 + 30 + 20 = 100 km.
• Budgeting for a School Event: When planning a school event, you can use a polynomial to calculate the total expenses. For example, if the venue costs ₹500, refreshments ₹300, and decorations ₹200, the total cost is:Total Cost = 500 + 300 + 200 = ₹1000.
• Estimating Time for Homework: If you have multiple homework assignments, you can add the time required for each to estimate the total time needed. For example, if one assignment takes 30 minutes, another 45 minutes, and another 60 minutes, the total time is: Total Time = 30 + 45 + 60 = 135 minutes.