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## Add polynomials by combining like terms

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As the students rounded the corner on Fifth Street, they spotted a peculiar looking building. It was in the shape of a pyramid. Mrs. Meery, the math teacher, asked her students the following question.

“A pyramid-shaped building has rectangular floors that get increasingly smaller as you go higher up in the building. If the 87th floor has a length of \begin{align*}6x + 16\end{align*} and a width of 28, and each floor’s length and width decrease by 4 as you ascend, find the total area of the 87th, 88th, and 89th floor.”

In this concept, you will learn to add polynomials.

A polynomial is an algebraic expression that shows the sum of monomials. In this concept you are going to add polynomials, but first let’s review how to add whole numbers with many digits.

Add the numbers 5026 and 3210.

You might choose to add it like this.

\begin{align*}5026 \\ \underline{+3210}\\ 8236\end{align*}

If you think about it, you might notice that the same addition could be thought of in this way.

\begin{align*}& \qquad \qquad \qquad \ \ \text{thousands} \quad \qquad \text{hundreds} \quad \qquad \ \text{tens} \quad \qquad \text{ones} \\ & \quad 5026 \quad \rightarrow \qquad \ \ 5000 \qquad \qquad \qquad \qquad \qquad \ \quad 20 \qquad \qquad \ \ 6\\ & \underline{+ \ 3210 \quad \rightarrow \quad \ + 3000 \qquad \qquad \ \ \ 200 \qquad \ \quad \quad \ \ \ 10 \qquad \qquad \qquad}\\ & \quad 8236 \quad \leftarrow \qquad \ \ 8000 \quad \quad \quad \ \ + 200 \quad \quad \quad \ \ \ + 30 \quad \ \quad \ \ \ \ + 6\end{align*}

Each of the similar places has been lined up vertically (one on top of the other) so that 3000 is beneath 5000 in the thousands place and 10 is beneath 20 in the tens place. Also, 200 is by itself because the first number had no digits in the hundreds place. Likewise, 6 is by itself because the second number had no digits in the ones place. Although this is not a practical way of writing a simple addition problem, it does demonstrate the technique you can use to add polynomials. Polynomials can be added in the same manner as we added 5026 and 3210.

You also need to know how to identify like terms. Like terms have exactly the same variable(s) to exactly the same power(s). When terms are alike, you can combine them by adding their coefficients.

For example:

\begin{align*}5x^3+9x^3=14x^3\end{align*}

Let’s look at an example.

Add the polynomials \begin{align*}(7x^2+9x-5)\end{align*} and \begin{align*}(6x^2+3x+10)\end{align*}.

First, line up the like terms so that you can add them vertically.

\begin{align*}& \ \ \ \ 7x^2+9x-5 \quad \rightarrow \quad 7x^2 \quad + \quad 9x \quad + \quad -5 \\ & \underline{+ \ 6x^2+3x+10 \ \ \rightarrow + \ 6x^2 \quad + \quad 3x \quad + \quad \ 10 \;} \\ & \ \ 13x^2+12x+5 \ \ \leftarrow \ \ \ 13x^2 \ \ \ + \quad 12x \ \ \ + \quad \ 5\end{align*}

Each of the like terms was aligned vertically, one on top of the other. Notice that the negative sign on -5 was kept with the number 5. Be careful when you add the integers.

A second method for adding polynomials is horizontally—in a single line. Just as you might add \begin{align*}6 + 19 = 25\end{align*} without placing them one on top of the other, polynomials can also be added horizontally.

Let’s look at an example.

Add the polynomials \begin{align*}(7x^2+3x-11)\end{align*} and \begin{align*}(3x^2-9x+5)\end{align*}.

First, rewrite the polynomials without parentheses. The polynomial can be rewritten without parentheses because the parentheses serve only to show the separation of the polynomials.

\begin{align*}(7x^2+3x-11)+(3x^2-9x+5)=7x^2+3x-11+3x^2-9x+5\end{align*}

Next, combine like terms.

\begin{align*}\begin{array}{rcl} (7x^2+3x-11)+(3x^2-9x+5) &=& {\color{blue}7x^2}+{\color{red}3x}-11+{\color{blue}3x^2}{\color{red}-9x}+5 \\ &=& {\color{blue}10x^2}{\color{red}-6x}-6 \end{array}\end{align*}

The answer is \begin{align*}10x^2-6x-6\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Mrs. Meery and the pyramids.

First, write the expression for finding the area of the 87th, 88th, and 89th floor.

\begin{align*}87^{th} \ \text{floor} & : \text{Area} = 28(6x + 16) \\ 88^{th} \ \text{floor} & : \text{Area} = (28 - 4)(6x + 16 - 4) \\ 89^{th} \ \text{floor} & : \text{Area} = (28 - 4 - 4)(6x + 16 - 4 - 4) \end{align*}

Next, find the areas for the three floors.

\begin{align*}87^{th} \ \text{floor} & : \text{Area} = 28(6x + 16) \\ & : \text{Area} = 168x + 448 \\ \\ 88^{th} \ \text{floor} & : \text{Area} = (28 - 4)(6x + 16 - 4) \\ & : \text{Area} = 24(6x + 12) \\ & : \text{Area} = 144x + 288 \\ \\ 89^{th} \ \text{floor} & : \text{Area} = (28 - 4 - 4)(6x + 16 - 4 - 4) \\ & : \text{Area} = 20(6x + 8) \\ & : \text{Area}= 120x + 160 \end{align*}

Then, find the total area for the three floors.

\begin{align*}\text{Total area} &= A_{87^{th} \ \text{floor}}+A_{88^{th} \ \text{floor}}+A_{89^{th} \ \text{floor}}\\ &= ({\color{blue}168x}+ 448) + ({\color{blue}144x }+ 288) + ({\color{blue}120x} + 160) \\ &={\color{blue}432x }+ 896\end{align*}

The answer is \begin{align*}432x+896\end{align*}.

The total area for the 87th, 88th, and 89th floors \begin{align*}432x+896\end{align*} is units squared.

#### Example 2

Add the polynomials \begin{align*}(-2x^3+9x^2-3)\end{align*} and \begin{align*}(8x^5+5x-14)\end{align*}.

First, rewrite the polynomials without parentheses.

\begin{align*}(-2x^3+9x^2-3)+(8x^5+5x-14)=-2x^3+9x^2-3+8x^5+5x-14\end{align*}

Next, combine like terms.

\begin{align*} \begin{array}{rcl} (-2x^3+9x^2-3)+(8x^5+5x-14) &=& {\color{pink}-2x^3}+{\color{blue}9x^2}-3+{\color{blue}8x^2}+{\color{red}5x}-14\\ &=& {\color{pink}-2x^3}+{\color{blue}17x^2}{\color{red}+5x}-17 \end{array}\end{align*}

The answer is \begin{align*}-2x^3+17x^2+5x-17\end{align*}.

#### Example 3

Add the polynomials \begin{align*}(4x^2+7x-2)\end{align*} and \begin{align*}(3x^2+2x-1)\end{align*}.

First, rewrite the polynomials without parentheses.

\begin{align*}(4x^2+7x-2)+(3x^2+2x-1)=4x^2+7x-2+3x^2+2x-1\end{align*}

Next, combine like terms.

\begin{align*}\begin{array}{rcl} (4x^2+7x-2)+(3x^2+2x-1) &=& {\color{blue}4x^2}+{\color{red}7x}-2+{\color{blue}3x^2}{\color{red}+2x}-1 \\ &=& {\color{blue}7x^2}{\color{red}+9x}-3 \end{array}\end{align*}

The answer is \begin{align*}7x^2+9x-3\end{align*}.

#### Example 4

Add the polynomials \begin{align*}(-4x^2+7x-2)\end{align*} and \begin{align*}(-7x^2+3x-17)\end{align*}.

First, rewrite the polynomials without parentheses.

\begin{align*}(-4x^2+7x-2)+(-7x^2+3x-17)=-4x^2+7x-2-7x^2+3x-17\end{align*}

Next, combine like terms.

\begin{align*}\begin{array}{rcl} (-4x^2+7x-2)+(-7x^2+3x-17) &=& {\color{blue}-4x^2}+{\color{red}7x}-2+{\color{blue}-7x^2}{\color{red}+3x}-17 \\ &=& {\color{blue}-11x^2}{\color{red}+10x}-19 \end{array}\end{align*}

The answer is \begin{align*}-11x^2+10x-19\end{align*}.

#### Example 5

Add the polynomials \begin{align*}(4xy+7x-2)\end{align*} and \begin{align*}(-19xy-17x-9)\end{align*}.

First, rewrite the polynomials without parentheses.

\begin{align*}(4xy+7x-2)+(-19xy-17x-9)=4xy+7x-2-19xy-17x-9\end{align*}

Next, combine like terms.

\begin{align*}\begin{array}{rcl} (4xy+7x-2)+(-19xy-17x-9) &=& {\color{blue}4xy}+{\color{red}7x}-2+{\color{blue}-19xy}{\color{red}-17x}-9 \\ &=& {\color{blue}-15x^2}{\color{red}-10x}-11 \end{array}\end{align*}

The answer is \begin{align*}15x^2-10x-11\end{align*}.

### Review

Add the following polynomials vertically. Be sure to align like terms.

1. \begin{align*}(4x^2+7x-2)+(3x-17)\end{align*}

2. \begin{align*}(-4x^4-x^3+8)+(-2x^3+5x+6)\end{align*}

3. \begin{align*}(10x^3-4x^2-2x+5)+(-x^2+9x-5)\end{align*}

4. \begin{align*}(6x^2+5x+9)+(4x^2+3x+6)\end{align*}

5. \begin{align*}(9x^2-3x+4)+(6x^2-9x+2)\end{align*}

6. \begin{align*}(3y^2+4x-9)+(-5y^2-6x+10)\end{align*}

7. \begin{align*}(14x^2+6x-2)+(9x-1)\end{align*}

8. \begin{align*}(-2x^2+7x-2)+(-3x^2-17)\end{align*}

9. \begin{align*}(9x^2+7x-2y)+(3x^2-x+9y)\end{align*}

10. \begin{align*}(4xy+7x-21)+(-12xy+4x-8)\end{align*}

11. \begin{align*}(11x^2+9x-2y)+(3x^2-8x-5y-2)\end{align*}

12. \begin{align*}(-3x-8)+(15x+5)\end{align*}

13. \begin{align*}(x^4+7x^3-2x+7)+(-8x^3+9x^2-4)\end{align*}

14. \begin{align*}(4x^2y-3x^2y^2+7xy)+(9x^2y^2-5xy+3x^2)\end{align*}

15. \begin{align*}(5xy-3x+19)+(4xy-9x-22)\end{align*}

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### Vocabulary Language: English

Area

Area is the space within the perimeter of a two-dimensional figure.

like terms

Terms are considered like terms if they are composed of the same variables with the same exponents on each variable.

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.