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# Addition and Subtraction of Polynomials

## Combining like terms in polynomial expressions

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Practice Addition and Subtraction of Polynomials
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What if you had two polynomials like $4x^2 - 5$ and $13x + 2$ ? How could you add and subtract them? After completing this Concept, you'll be able to perform addition and subtraction on polynomials like these.

### Try This

For more practice adding and subtracting polynomials, try playing the Battleship game at http://www.quia.com/ba/28820.html . (The problems get harder as you play; watch out for trick questions!)

### Guidance

To add two or more polynomials, write their sum and then simplify by combining like terms.

#### Example A

Add and simplify the resulting polynomials.

a) Add $3x^2-4x+7$ and $2x^3-4x^2-6x+5$

b) Add $x^2-2xy+y^2$ and $2y^2-3x^2$ and $10xy+y^3$

Solution

a) $& (3x^2-4x+7)+(2x^3-4x^2-6x+5)\\\text{Group like terms:} & = 2x^3+(3x^2-4x^2)+(-4x-6x)+(7+5)\\\text{Simplify:} & = 2x^3-x^2-10x+12$

b) $& (x^2-2xy+y^2)+(2y^2-3x^2)+(10xy+y^3)\\\text{Group like terms:} & = (x^2-3x^2)+(y^2+2y^2)+(-2xy+10xy)+y^3\\\text{Simplify:} & = -2x^2+3y^2+8xy+y^3$

To subtract one polynomial from another, add the opposite of each term of the polynomial you are subtracting.

#### Example B

a) Subtract $x^3-3x^2+8x+12$ from $4x^2+5x-9$

b) Subtract $5b^2-2a^2$ from $4a^2-8ab-9b^2$

Solution

a) $(4x^2+5x-9)-(x^3-3x^2+8x+12) & = (4x^2+5x-9)+(-x^3+3x^2-8x-12)\\\text{Group like terms:} & = -x^3+(4x^2+3x^2)+(5x-8x)+(-9-12)\\\text{Simplify:} & = -x^3+7x^2-3x-21$

b) $(4a^2-8ab-9b^2)-(5b^2-2a^2) & = (4a^2-8ab-9b^2)+(-5b^2+2a^2)\\\text{Group like terms:} & = (4a^2+2a^2)+(-9b^2-5b^2)-8ab\\\text{Simplify:} & = 6a^2-14b^2-8ab$

Note: An easy way to check your work after adding or subtracting polynomials is to substitute a convenient value in for the variable, and check that your answer and the problem both give the same value. For example, in part (b) above, if we let $a=2$ and $b=3$ , then we can check as follows:

$& \text{Given} && \text{Solution}\\& (4a^2-8ab-9b^2)-(5b^2-2a^2) && 6a^2-14b^2-8ab\\& (4(2)^2-8(2)(3)-9(3)^2)-(5(3)^2-2(2)^2) && 6(2)^2-14(3)^2-8(2)(3)\\& (4(4)-8(2)(3)-9(9))-(5(9)-2(4)) && 6(4)-14(9)-8(2)(3)\\& (-113)-37 && 24-126-48\\& -150 && -150$

Since both expressions evaluate to the same number when we substitute in arbitrary values for the variables, we can be reasonably sure that our answer is correct.

Note: When you use this method, do not choose 0 or 1 for checking since these can lead to common problems.

Problem Solving Using Addition or Subtraction of Polynomials

One way we can use polynomials is to find the area of a geometric figure.

#### Example C

Write a polynomial that represents the area of each figure shown.

a)

b)

c)

d)

Solution

a) This shape is formed by two squares and two rectangles.

$\text{The blue square has area} \ y \times y & = y^2.\\\text{The yellow square has area} \ x \times x & = x^2.\\\text{The pink rectangles each have area} \ x \times y & = xy.$

To find the total area of the figure we add all the separate areas:

$Total \ area &= y^2 + x^2 + xy + xy\\& = y^2 + x^2 + 2xy$

b) This shape is formed by two squares and one rectangle.

$\text{The yellow squares each have area} \ a \times a & = a^2.\\\text{The orange rectangle has area} \ 2a \times b & = 2ab.$

To find the total area of the figure we add all the separate areas:

$Total \ area & = a^2 + a^2 + 2ab\\& = 2a^2 + 2ab$

c) To find the area of the green region we find the area of the big square and subtract the area of the little square.

$\text{The big square has area}: y \times y & = y^2.\\\text{The little square has area}: x \times x & = x^2.\\Area \ of \ the \ green \ region & = y^2 - x^2$

d) To find the area of the figure we can find the area of the big rectangle and add the areas of the pink squares.

$\text{The pink squares each have area} \ a \times a & = a^2.\\\text{The blue rectangle has area} \ 3a \times a & = 3a^2.$

To find the total area of the figure we add all the separate areas:

$Total \ area = a^2 + a^2 + a^2 + 3a^2 = 6a^2$

Another way to find this area is to find the area of the big square and subtract the areas of the three yellow squares:

$\text{The big square has area} \ 3a \times 3a & = 9a^2.\\\text{The yellow squares each have area} \ a \times a & = a^2.$

To find the total area of the figure we subtract:

$Area & = 9a^2 - (a^2 + a^2 + a^2)\\& = 9a^2 - 3a^2 \\& = 6a^2$

Watch this video for help with the Examples above.

### Vocabulary

• A polynomial is an expression made with constants, variables, and positive integer exponents of the variables.
• In a polynomial, the number appearing in each term in front of the variables is called the coefficient.
• In a polynomial, the number appearing all by itself without a variable is called the constant.
• Like terms are terms in the polynomial that have the same variable(s) with the same exponents, but they can have different coefficients.

### Guided Practice

Subtract $4t^2+7t^3-3t-5$ from $6t+3-5t^3+9t^2$ .

Solution:

When subtracting polynomials, we have to remember to subtract each term. If the term is already negative, subtracting a negative term is the same thing as adding:

$6t+3-5t^3+9t^2-(4t^2+7t^3-3t-5)&= \\6t+3-5t^3+9t^2-(4t^2)-(7t^3)-(-3t)-(-5)&=\\6t+3-5t^3+9t^2-4t^2-7t^3+3t+5&=\\(6t+3t)+(3+5)+(-5t^3-7t^3)+(9t^2-4t^2)&=\\9t+8-12t^3+5t^2&=\\-12t^3+5t^2+9t+8\\$

The final answer is in standard form.

### Explore More

1. $(x+8)+(-3x-5)$
2. $(-2x^2+4x-12)+(7x+x^2)$
3. $(2a^2b-2a+9)+(5a^2b-4b+5)$
4. $(6.9a^2-2.3b^2+2ab)+(3.1a-2.5b^2+b)$
5. $\left ( \frac{3}{5}x^2-\frac{1}{4}x+4 \right )+ \left ( \frac{1}{10}x^2 + \frac{1}{2}x-2\frac{1}{5} \right )$

Subtract and simplify.

1. $(-t+5t^2)-(5t^2+2t-9)$
2. $(-y^2+4y-5)-(5y^2+2y+7)$
3. $(-5m^2-m)-(3m^2+4m-5)$
4. $(2a^2b-3ab^2+5a^2b^2)-(2a^2b^2+4a^2b-5b^2)$
5. $(3.5x^2y-6xy+4x)-(1.2x^2y-xy+2y-3)$

Find the area of the following figures.

### Vocabulary Language: English

distributive property

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.
Polynomial

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.