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# 12.6: Subtracting Polynomials

Difficulty Level: At Grade Created by: CK-12
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Practice Addition and Subtraction of Polynomials

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Have you ever thought about the area of a pool? Take a look at this dilemma.

A concrete walkway surrounds a rectangular swimming pool. In order to know how much weather treatment to buy, the owner must know how many square units of concrete he has. The walkway is 5 feet wide on all sides. The swimming pool has a length of 7x\begin{align*}7x\end{align*} and a width of 14 feet. How many square units concrete does he have?

In order to find the area of the concrete, we must find the area of the large rectangle and then subtract the area of the swimming pool.

Do you know how to do this? To accomplish this task, you will need to understand how to subtract polynomials. You will learn how to do this in this Concept.

### Guidance

A polynomial is an algebraic expression that shows the sum of monomials.

Since the prefix mono means one, a monomial is a single piece or term. The prefix poly means many. So the word polynomial refers to one or more than one term in an expression. The relationship between these terms may be sums or difference.

Polynomials: x2+53x8+4x57a2+9b4b3+6\begin{align*}x^2+ 5 \qquad 3x-8+4x^5 \qquad -7a^2+9b-4b^3+6\end{align*}

We call an expression with a single term a monomial, an expression with two terms is a binomial, and an expression with three terms is a trinomial. An expression with more than three terms is named simply by its number of terms—“five-term polynomial.”

Just like we could add polynomials, we can subtract them too. We can perform this operation both vertically and horizontally. Let’s start with vertically.

When we subtract polynomials, we can use a similar procedure as with addition—we can subtract vertically. However, remember that subtraction is the same as “adding the opposite.” In other words, 58\begin{align*}5 - 8\end{align*} is the same as 5+(8)\begin{align*}5 + (-8)\end{align*}. We can add the opposite of 8 instead of subtracting 8. We will use the same idea with polynomials.

Remember that subtracting is the same as adding the opposite. Write this down in your notebook.

Take a look at this one.

(9x2+4x7)(2x2+6x4)\begin{align*}(9x^2+4x-7)-(2x^2+6x-4)\end{align*}

Set up the problem vertically by aligning the like terms.

(9x2+4x7)  9x2+  4x+7Lines up like terms.(2x2+6x4)2x2+6x+ 4 Add the opposite.  7x22x3    7x2+ 2x  +3Combine lilke terms.\begin{align*}& \quad (9x^2+4x-7) \quad \rightarrow \ \ \quad 9x^2 \quad + \ \ \quad 4x \quad + \quad -7 \quad \text{Lines up like terms.} \\ & \underline{-(2x^2+6x-4) \quad \rightarrow \quad -2x^2 \quad + \quad -6x \quad + \ \quad 4 \quad \ \text{Add the opposite.} \quad \ \ } \\ & \quad 7x^2-2x-3 \quad \ \ \leftarrow \quad \ \ 7x^2 \quad + \ \quad -2x \ \ + \quad -3 \quad \text{Combine lilke terms.}\end{align*}

When you add the opposite, the sign changes on each of the terms in the subtracted polynomial. Inside the parentheses, the coefficient of 2x2\begin{align*}2x^2\end{align*} is positive. But when you add the opposite, the sign changes to negative, or 2x2\begin{align*}-2x^2\end{align*}. We also changed the sign on the 6x\begin{align*}6x\end{align*} to 6x\begin{align*}-6x\end{align*} and the -4 to 4.

Now we can look at subtracting polynomials horizontally.

When we added polynomials, we used two methods—adding vertically and adding horizontally. You just learned to subtract polynomials vertically. As you have guessed, we can also subtract polynomials horizontally. First we will review the distributive property.

The distributive property: For all real numbers a,b,\begin{align*}a, b,\end{align*} and c, a(b+c)=ab+ac\begin{align*}c, \ a(b + c) = ab + ac\end{align*}.

5(3x+7)4(2y7)2(9x+3)3(2y4)=15x+35=8y28=18x6=6y+12\begin{align*}5(3x+7)&=15x+35 \\ 4(2y-7)&=8y-28 \\ -2(9x+3)&=-18x-6 \\ -3 (-2y-4)&=6y+12\end{align*}

Remember to be careful with negative signs when using distributive property.

Now, let us remember that coefficients are the numerical factors of variables. The coefficient of 3x\begin{align*}3x\end{align*} is 3. The coefficient of 9x2\begin{align*}9x^2\end{align*} is 9. When we see the term x\begin{align*}-x\end{align*}, the coefficient is -1. Although you could write 1x\begin{align*}-1x\end{align*}, we normally do not because the 1 is considered unnecessary. How does this relate to the distributive property? The negative sign could be in front of the parentheses, like this: (3x2)\begin{align*}-(3x - 2)\end{align*}. This is similar to x\begin{align*}-x\end{align*} where the coefficient is the unwritten -1. Just like you could write 1x\begin{align*}-1x\end{align*}, you could also write 1(3x2)\begin{align*}-1(3x - 2)\end{align*}. The distributive property is now more apparent in that each term will now be multiplied by -1.

Take a look at this one.

(7x+5)=1(7x+5)=7x5\begin{align*}& -(7x+5)=-1(7x+5)=-7x-5\end{align*}

Then take a look at this one.

(x23x+14)=1(x23x+14)=x2+3x14\begin{align*}& -(x^2-3x+14)=-1(x^2-3x+14)=-x^2+3x-14 \\ & \qquad \qquad \quad \uparrow \end{align*}

Here you can insert the -1 and then multiply. As with adding the opposite, the sign changes on each of the terms in the polynomial.

We can now use this method to subtract polynomials horizontally. First we’ll distribute the negative sign to each of the terms in the subtracted polynomial and then we will combine like terms just as we did when we added polynomials.

Here is another one.

(5x+3)(2x8)=(5x+3)1(2x8)=5x+32x+8=3x+11\begin{align*}& (5x+3)-(2x-8)\\ &=(5x+3)-1(2x-8) \\ &=5x+3-2x+8 \\ &=3x+11 \end{align*}

It may seem that way, but if you go step by step and remember that subtracting is adding the opposite, then you will be able to subtract polynomials vertically and horizontally.

Subtract the following polynomials.

#### Example A

(8x2+4x7)(2x2+9x+3)\begin{align*}(8x^2+4x-7)-(2x^2+9x+3)\end{align*}

Solution: 6x25x10\begin{align*}6x^2-5x-10\end{align*}

#### Example B

(10xy+4x7)(3x4)\begin{align*}(10xy+4x-7)-(3x-4)\end{align*}

Solution: 10xy+x3\begin{align*}10xy+x-3\end{align*}

#### Example C

(14x2+8x7y+1)(2x2+2x4y+2)\begin{align*}(14x^2+8x-7y+1)-(2x^2+2x-4y+2)\end{align*}

Solution: 12x2+6x3y1\begin{align*}12x^2+6x-3y-1\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

The length of the large rectangle measures 7x+5+5\begin{align*}7x + 5 + 5\end{align*}. Its width measures 14+5+5\begin{align*}14 + 5 + 5\end{align*}.

So its area will be (7x+5+5)(14+5+5)\begin{align*}(7x + 5 + 5) \cdot (14 + 5 + 5)\end{align*}

The area of the swimming pool will be its length times its width or 7x14\begin{align*}7x \cdot 14\end{align*}.

Area of the swimming pool is 98x\begin{align*}98x\end{align*}.

To find the area of the concrete, subtract the area of the swimming pool from the total area:

(168x+240)98x=70x+240\begin{align*} &(168x + 240)-98x \\ &=70x + 240 \end{align*}

### Vocabulary

Polynomial
one or more terms in an expression, often referred to specifically in situations where there are more than three terms.
Like Terms
terms that have the same variable and power.
Area
the space inside an object or area. It is measured in square units.

### Guided Practice

Here is one for you to try on your own.

(7x3+3x2x+4)(6x2+9)\begin{align*}(-7x^3+3x^2-x+4)-(-6x^2+9)\end{align*}

Solution

(7x3+3x2x+4)  7x3+  3x2+x+4 Lines up like terms.(6x2+9)  6x2+  +9   Add the opposite.7x3+9x2x5 7x3+ 9x2+x+5 Combine lilke terms.\begin{align*}& \quad (-7x^3+3x^2-x+4) \quad \rightarrow \ \ \quad -7x^3 \quad + \ \ \quad 3x^2 \quad + \quad -x \quad + \quad \quad 4 \quad \ \text{Lines up like terms.} \\ & \underline{\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; -(-6x^2+9) \quad \rightarrow \quad \quad \quad \quad \quad \quad \quad \ \ 6x^2 \quad + \ \quad \quad \quad \ + \quad -9 \ \ \ \text{Add the opposite.}\; \; \; \; \; \;} \\ & \quad -7x^3+9x^2-x-5 \quad \ \leftarrow \quad \quad -7x^3 \quad + \quad \ 9x^2 \quad + \quad -x \quad + \quad -5 \quad \ \text{Combine lilke terms.}\end{align*}

### Practice

Subtract the following polynomials vertically.

1. (6x2+5x)(3x214x+2)\begin{align*}(6x^2+5x)-(3x^2-14x+2)\end{align*}
2. (3x2+5x+3)(2x2x+4)\begin{align*}(3x^2+5x+3)-(2x^2-x+4)\end{align*}
3. (5xy+5x+3)(12xy4x8)\begin{align*}(5xy+5x+3)-(12xy-4x-8)\end{align*}
4. (5y2+5y2)(3y26y+5)\begin{align*}(5y^2+5y-2)-(3y^2-6y+5)\end{align*}
5. (8x+5y+1)(9x+2y+5)\begin{align*}(8x+5y+1)-(9x+2y+5)\end{align*}
6. (7x2+x3)(3x2+3x+4)\begin{align*}(7x^2+x-3)-(3x^2+3x+4)\end{align*}
7. (8x+5y+4)(3x9y5)\begin{align*}(8x+5y+4)-(3x-9y-5)\end{align*}
8. (18x3+2x2+8x+2)(3x24x9)\begin{align*}(18x^3+2x^2+8x+2)-(3x^2-4x-9)\end{align*}
9. (8x+9y20)(3x14)\begin{align*}(8x+9y-20)-(3x-14)\end{align*}
10. (16x2+5x3y+7)(3x14y+10)\begin{align*}(16x^2+5x-3y+7)-(3x-14y+10)\end{align*}
11. (18x2+5xy6x+21)(3x214xy9x+1)\begin{align*}(18x^2+5xy-6x+21)-(3x^2-14xy-9x+1)\end{align*}
12. (7y3+4y23y1)(y3+6y24)\begin{align*}(7y^3+4y^2-3y-1)-(y^3+6y^2-4)\end{align*}

Subtract the following polynomials horizontally.

1. \begin{align*}(m^2+17m-11)-(3m^2+8m+12)\end{align*}
2. \begin{align*}(z^2+3z)-(3z^2+7z+16)-(4z-13)\end{align*}
3. \begin{align*}(5x^2+3xy)-(3x^2+7xy+6)-(4xy-13)\end{align*}

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### Vocabulary Language: English

TermDefinition
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.

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