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# Adding and Subtracting Polynomials – Explanation & Examples

**A polynomial is an expression that contain variables and coefficients**.

For example, ax + b, 2x^{2} – 3x + 9 and x^{4} – 16 are polynomials.

**The word “polynomial”** is derived from the words “**poly**” and “**nomial**”, which means many and terms respectively. A polynomial can have variables, constants and exponents, but an expression is not a polynomial if the variable is in the denominator, like 2/x + 3, 9xy^{-2}, etc.

Like the numbers, they can undergo same type of operations. The operation of adding and subtracting polynomials is as easy as pie. You only need to be familiar with how to combine like terms and also the order of operations within the question. Before we can begin, let’s us recall, what like terms are.

In mathematics, like terms are terms that contain identical variables and exponents, regardless of their coefficients. An expression can be simplified by adding or subtracting depending on the signs before the terms.

**For example**, 7xy + 6y + 6xy is a polynomial who’s like terms are 7xy and 6xy. Therefore, this polynomial can be simplified by combining like terms as 7xy +6xy +6y = 13xy + y. When combining like terms, we only add or subtract the coefficients of the identical variables.

On the other hand, unlike terms are terms which are not identical in terms of either variables or exponents.

**For example**, an expression 4x + 9y^{2}, contain unlike terms because variable x and y are different and are not raised to the same power.

## How to Add Polynomials?

Addition of polynomials involves arranging the like terms together and summing them up.

You can perform the operation by arranging the polynomials either vertically or horizontally. Whichever method you will use, the final answer will remain the same.

*Example 1*

Add the following polynomials:

5x + 3y, 4x – 4y + z and -3x + 5y + 2z

__Solution__

The first step is to combine the polynomials by the addition operators.

= (5x + 3y) + (4x – 4y + z) + (-3x + 5y + 2z)

**= **5x + 3y + 4x – 4y + z – 3x + 5y + 2z

Now arrange the like terms together and add

= 5x + 4x – 3x + 3y – 4y + 5y + z + 2z

= 6x + 4y + 3z

*Example 2*

Add: 3a^{2} + ab – b^{2}, -a^{2} + 2ab + 3b^{2} and 3a^{2} – 10ab + 4b^{2}

__Solution__

Combine the polynomials by the addition operators.

= (3a^{2} + ab – b^{2}) + (-a^{2} + 2ab + 3b^{2}) + (3a^{2} – 10ab + 4b^{2})

= 3a^{2} + ab – b^{2} – a^{2} + 2ab + 3b^{2} + 3a^{2} – 10ab + 4b^{2}

Arrange the like terms together and then add

= 3a^{2} – a^{2} + 3a^{2} + ab + 2ab – 10ab – b^{2} + 3b^{2} + 4b^{2}

= 5a^{2} – 7ab + 6b^{2}

*Example 3*

Add the polynomials below.

15x^{3} – 6x – 23, 3x^{3} – 5x^{2} + 8x + 10, -8x^{3} + 2x^{2} – 7x and 9x^{2} – 4x + 15

__Solution__

Combine the polynomials:

(15x^{3} – 6x – 23) + (3x^{3} – 5x^{2} + 8x + 10) + (-8x^{3} + 2x^{2} – 7x) + (9x^{2} – 4x + 15)

Arrange the like terms together and add;

= (15x^{3} + 3x^{3} – 8x^{3}) + (– 5x^{2} + 2x^{2 }+ 9x^{2}) + (– 6x + 8x – 7x– 4x) + (– 23 + 10 +15)

= 10x^{3} + 6x^{2 }– 9x + 2

*Example 4*

Add: (3x^{3} – 5x + 9) + (6x^{3} + 8x – 7)

__Solution__

If the problem has parentheses, remove them by applying the distributive property of multiplication.

(3x^{3} – 5x + 9) + (6x^{3} + 8x – 7) ⟹ 3x^{3} – 5x + 9 + 6x^{3} + 8x – 7

Arrange the like terms together and add;

⟹ 3x^{3} + 6x^{3} + (-5x) + 8x + 9 + (-7)

= 9x^{3} + 3x + 2

*Example 5*

Add the following polynomial:

(2x^{2 +} 5x + 7) + (3x^{2 }−2x + 5)

__Solution__

Apply the commutative property to group like terms.

⟹ (2x^{2 }+ 3x^{2}) + (5x −2x) + (7 + 5)

Now use the distributive property.

⟹ (2 + 3) x^{2 }+ (5−2) x + (7 + 5)

=5x^{2 }+ 3x + 12

## How to Subtract Polynomials?

Polynomials can be subtracted by either method. You can subtract by arranging the polynomials in a horizontal or vertical form.

*To subtract polynomials horizontally, here are the steps:*

- First enclose the subtracting polynomial into brackets such that the minus sign is prefixed.
- Now remove the brackets by accordingly manipulating the sign in each term of polynomial i.e. (– changes to + and vice versa).
- Arrange the like terms together and add the likes together. We add instead of subtracting because the minus sign was changed when removing the parentheses.

** NOTE**: The polynomial or expression that comes before the word “from” is the subtracting quantity.

*Example 6*

Subtract the following polynomial 2x – 5y + 3z from 5x + 9y – 2z.

__Solution__

Enclose the subtracting polynomial and place a negative sign in front of the parentheses.

⟹ 5x + 9y – 2z – (2x – 5y + 3z)

Now open the parentheses by manipulating the signs

= 5x + 9y – 2z – 2x + 5y – 3z

= 5x – 2x + 9y + 5y – 2z – 3z

= 3x + 14y – 5z

*Example 7*

Subtract the polynomials below:

-6x^{2} – 8y^{3} + 15z from x^{2} – y^{3} + z.

__Solution__

Enclose the subtracting polynomial.

⟹ x^{2} – y^{3} + z – (-6x^{2} – 8y^{3} + 15z)

Remove the parentheses by changing the operators within the parentheses

= x^{2} – y^{3} + z + 6x^{2} + 8y^{3} – 15z

Arrange the like terms together.

= x^{2} + 6x^{2} – y^{3} + 8y^{3} + z – 15z

= 7x^{2} + 7y^{3} – 14z

*Example 8*

Subtract: 3x^{3} + 5x^{2} – 7x + 10 from 6x^{3} – 8x^{2} + x + 10

__Solution__

Enclose the subtracting trinomial in parentheses

⟹ 6x^{3} – 8x^{2} + x + 10 – (3x^{3} + 5x^{2} – 7x + 10)

Remove the parentheses by changing the sign of each term inside the parentheses

⟹ 6x^{3} – 8x^{2} + x + 10 – 3x^{3} – 5x^{2} + 7x – 10)

Arrange the like terms and add to get;

= 3x^{3} – 13x^{2} + 8x

*Practice Questions*

- Subtract (5x
^{3}– 7x^{2}– 8) – (4x^{2}+ 5x – 6) - Add 4x
^{3}– 9x + 3 and 5x^{2}– 4x + 7. - Subtract 4x
^{2}– 7x + 5 from 3x^{2}– 2x + 6 - Solve (–3x
^{2}+ 9xy – 5y^{2}) – (4x^{2}+ 7xy – 8y^{2}) - Determine the expression that should be subtracted from 3x + 5y + 9 to get – 2x + 3y + 15.
- The sum of two polynomials is 3x
^{2}+ 2xy – y^{2}. Determine the other polynomial if one of them is 2x^{2}+ 3y^{2}. - How much is 3a + 5b – 4c greater than 5a + 6b – 3c
- How much is –pq + qr – rp less than qr – rp + pq
- Take a – 2b – c from the sum of a + b – 3c and 3a – b + c
- By how much must 2p
^{2}+ q^{2 }incremented to give 5p^{2}– 3q^{2}?

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