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Adding and Subtracting Polynomials with Solved Examples

adding and subtracting polynomials

Have you wondered how you can add, subtract and multiply polynomials? Well, that’s easy and can help you uplift your grades in high school. Polynomials are algebraic expressions that consist of variables and coefficients. Polynomials can be classified into two parts: Poly meaning many and Nominal meaning terms. Polynomials are algebraic expressions made up of variables, constants, and exponents combined using mathematical operations, including addition, subtraction, multiplication, and division.

P(x) denotes a polynomial function, where x is the variable:

P(x) = x²+2x+10

We work with like terms to perform basic arithmetic operations like adding and subtracting polynomials.

Like Terms

As the name suggests, these are terms that are “like” each other. Like Terms are terms with the same variables. For example, for two polynomial functions P(x) and R(x):

P(x) = x²+2x+10

R(x)= 3x²+5x+12  

Here x² and 3x²; 2x and 5x, are Like Terms. 

Adding Polynomials

Adding Polynomials consists of two ways:

  • Horizontal Way of Adding

The horizontal way of adding polynomials is the same as the vertical way but the difference is just that the like terms of the polynomials are sorted and arranged in columns.

For example for adding two polynomials P(x)=x²+2x+10 and R(x)= 3x²+5x+12 : 

= (x²+2x+10) + (3x²+5x+12)

= (x²+3x²) + (2x+5x) + (12+10)

= (4x²) + (7x) + (22)

= 4x²+7x+22

  • Vertical Way of Adding

To add polynomials vertically, add all of the like terms of the corresponding polynomials in the following way:-

Step 1: Organize every polynomial in decreasing order of degree, starting with the term with the greatest degree and working your way down.

Step 2: Sort the terms that are alike, i.e, like terms whose variables are the same.

Step 3: Add the polynomials by placing them one above the other(vertically).

For example for adding two polynomials P(x)=x²+2x+10 and R(x)= 3x²+5x+12 :-

 x²+2x+10

 3x²+5x+12

——————

 4x²+7x+22        {Sum of P(x) and R(x)}

Subtracting Polynomials

Subtracting is exactly like adding polynomials just with added few steps of sign conversion.

  • Horizontal Way of Subtracting

Step 1: Put the subtracting polynomial in brackets so that the negative sign is at the beginning.

Step 2: Reverse the sign of each term we’re subtracting: “+” becomes “-“, and “-” becomes “+”.

Step 3: Sort the like terms together and add them together.

For example, subtracting polynomial P(x)=x²+2x+10 from R(x)= 3x²+5x+12 :

(x²+2x+10) – (3x²+5x+12)

= (x²+2x+10) + (-3x²-5x-12)

= (x²-3x²) + (2x-5x) + (10-12)

= (-2x²) + (-3x) + (-2)

= -2x²-3x-2

  • Vertical Way of Subtracting

Simply add all of the similar terms of the related polynomials after altering the sign to subtract polynomials vertically in the following way:-

Step 1: Organize every polynomial in decreasing order of degree, starting with the term with the greatest degree and working your way down.

Step 2: Reverse the sign of each term we’re subtracting: “+” becomes “-“, and “-” becomes “+”.

Step 3: Sort the terms that are alike, i.e, Like terms whose variables are the same, and simply add the polynomials by placing them one above the other(vertically).

For example for adding two polynomials P(x)=x²+2x+10 and R(x)= 3x²+5x+12 :-

x² + 2x + 10

– 3x² – 5x – 12

—————————

-2x² -3x -2        {Sum of P(x) and R(x)}

Adding and Subtracting Polynomials Quizlet

Q. Find the Sum of Polynomials −2x²−4x−2 and 3x²+5x+12.

= (−2x²−4x−2) + (3x²+5x+12)

= (-2x²+3x²) + (-4x+5x) + (-2+12)

= (x²) + (x) + (10)

= x²+x+10

Q. Simplify –4x + 7 – (5x – 3)

–4x + 7 – (5x – 3)

= –4x + 7 – 5x + 3

= –9x + 10

Q. Find the Sum of Polynomials pq+qr-rp and 2pq-qr+rp.

= (pq+qr-rp) + (2pq-qr+rp)

= (pq+2pq) + (qr-qr) + (-rp+rp)

= (3pq) + (0) + (0)

= 3pq

Q. Find the Difference of Polynomials 5y²+2xy−9 and 2y²+2xy-3.

      5y² + 2xy – 9

    – 2y² – 2xy + 3

—————————

 3y² – 6

Q. The sum of two polynomials is 6t²+7t+6.  Determine the other polynomial if one of them is 4t²-2. 

Let the other term be x.

Therefore, x + (4t²-2) = 6t²+7t+6

Or x =  6t²+7t+6 – (4t²-2)

        =  6t²+7t+6 -4t²+2

        =  (6t²-4t²)+7t+(6+2)

        = 2t²+7t+8                                                 (other polynomial)

Q. Find the Sum of Polynomials a+2 and a-2 and -2a+1.

= (a+2) + (a-2) + (-2a+1)

= (a+a-2a) + (2-2+1) 

= (0) + (1) 

= 1

Q. Find the Difference of Polynomials −12x²−14x−12 and x²+x+2.

(−12x²−14x−12) – (x²+x+2)

= (−12x²−14x−12) + (-x²-x-2)

= (-12x²-x²) + (-14x-x) + (-12-2)

= (-13x²) + (-15x) + (-14)

= -13x²-15x-14

Q. Find the Sum of Polynomials 2p²+q²  and 5p²-3q².

2p²+q²

 5p²-3q²

——————

  7p²-2q²

Q. How much greater is 4x²+4xy-12 than x²+1?

Let z be the value of how much greater 4x²+4xy-12 than x²+1 is,

Therefore, z + (x²+1) = 4x²+4xy-12

Or z = 4x²+4xy-12 – (x²+1)

z  = 4x²+4xy-12 – x²-1

= (4x²-x²)+4xy-12-1

= 3x² +4xy -13

Q. Find the Sum of Polynomials 5x²+4x−2 and 3x²+5x+12 and 12x²-4.

5x²+4x-2

3x²+5x+12

12x²+0-4       

——————

 20x²+9x+6

Q. Find the Difference of Polynomials 12x³−2x²−4x−2 and 5x³+3x²+5x+12.

(12x³-2x²-4x-2) – (5x³+3x²+5x+12)

= (12x³-2x²-4x-2) + (-5x³-3x²-5x-12)

=(12x³-5x³ )+ (-2x²-3x²) + (-4x-5x) + (-2-12)

= (7x³) + (-5x²) + (-9x) + (-14)

= 7x³-5x²-9x-14

Q. Find the Sum of Polynomials -x³−12x²+4x−32 and 2x³−2x²−4x+76.

 -x³−12x²+4x−32

 2x³−2x²−4x+76

——————

  x³ – 14x² + 44

Q. Find the difference of polynomials xyz-xy-z and -xy-z and xyz.

xyz – xy – z

– xy – z

-xyz

—————————

2xyz

Q. Find the sum of polynomials: x²+2x+10 and 3x²+5x+12.

= (x²+2x+10) + (3x²+5x+12)

= (x²+3x²) + (2x+5x) + (12+10)

= (4x²) + (7x) + (22)

= 4x²+7x+22

Q. Find the Difference of Polynomials x²+2x+10 and 3x²+5x+12.

(x²+2x+10) – (3x²+5x+12)

= (x²+2x+10) + (-3x²-5x-12)

= (x²-3x²) + (2x-5x) + (10-12)

= (-2x²) + (-3x) + (-2)

= -2x²-3x-2

Q. Find the Difference of Polynomials −2x²−4x−2 and 3x²+5x+12.

      -2x² – 4x – 2

    – 3x² – 5x – 12

—————————

 -5x² -9x -14

Q. Find the Sum of Polynomials 7y²−4x−8 and 8x²+1.

7y²-4x-8

 8x²+1

——————

  8x²+7y²-4x-7

Q. Find the sum of –7x³y + 4x²y – 2 and 4x³y + 1 – 8x²y²

–7x³y + 4x²y²– 2 + 4x³y + 1 – 8x²y²

= (–7x³y+4x³y) + (4x²y²– 8x²y²) – 2 + 1

= –3x³y – 4x²y² – 1

Q.  Find the perimeter of a triangle with side lengths: 8, (x-6), and (x+2).

The perimeter of a triangle = sum of all sides

= 8+ (x-6)+ (x+2)

=8+x-6+x+2

=x+x+8+2-6

= 2x+4

Q. Find the perimeter of a rectangle with a length of y²+12 and a width of 3y²-2y+2.

Perimeter of a rectangle =2( length + breadth)

= 2(y²+12 + 3y²-2y+2)

= 2y²+6y² -4y +24 +4

= 8y² -4y+ 28

Variables, constants, and exponents of a polynomial can all be subjected to the same operations. Adding, subtracting, and multiplying polynomials can easily be done. You need to know the rules for combining like terms and the order of operations within the query. Polynomial addition is straightforward. We add like terms while adding polynomials. Subtraction of polynomials is as simple and clear as the addition of polynomials, except for sign conversion. They can either be arranged vertically or horizontally when adding and subtracting polynomials.

 FAQs

Q1. What are polynomials?

Polynomials are algebraic formulas with variables and coefficients. Polynomials can be divided into two parts: Poly (many) and Nominal(terms). Polynomials are algebraic equations composed of variables, constants, and exponents that are combined via mathematical operations such as addition, subtraction, multiplication, and division.

Q2. What are the methods for adding and subtracting polynomials?

The addition or subtraction of polynomials is fairly straightforward. We do not need to have an adding and subtracting polynomials calculator. All we need to remember are a few steps. The polynomials can be placed vertically for complex equations to execute the addition and subtraction operations. We can also use the horizontal arrangement to complete the operation for easier calculations.

Q3. How are the like terms involved during the subtraction of polynomials?

Like terms include the same variables raised to the same power. The only difference is in the numerical coefficients. We combine like terms while conducting subtraction or addition. To make algebraic formulas easier to deal with, we combine like terms to shorten and simplify them.

For example, for two polynomial functions P(x) = x²+2x+10 and R(x)= 3x²+5x+12,  x² and 3x²; 2x and 5x, are Like Terms.

Q4. What’s the distinction between adding and subtracting polynomials?

When adding polynomials, like terms are added, and when subtracting polynomials, like terms are still added but after sign conversion wherein the polynomial which is being subtracted, all the signs of coefficient changes, i.e., “+” becomes “-“, and “-” becomes “+”.  Remember that the sign after addition or subtraction will always be of the variable with the greater value just like the basic rule.

Q5. In a polynomial, what are coefficients, constants, variables, degrees, and exponents?

For a polynomial  x² + 2x + 10 , following is how we describe:

Coefficient: 1,+2 

Constant: 10 

Degree: 2 (the highest exponent of variable x)

Exponents: Power raised to variable x, i.e. 2 and 1.

Q6. Can we consider a real number as a polynomial?

This is one of the questions asked in adding and subtracting polynomials quizlets. Let’s take 6 for instance,

6 is a real number and it can also be written in the form:-

0x²+x+6, for some variable x

6=0x²+x+6 Therefore, we can say that 6 is a polynomial.

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