7.1: Addition and Subtraction of Polynomials
You are going to build a rectangular garden in your back yard. The garden is 2 m more than 1.5 times as long as it is wide. Write an expression to show the area of the garden.
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Khan Academy Adding and Subtracting Polynomials 1
Guidance
The word polynomial comes from the Greek word poly meaning “many”. Polynomials are made up of one or more terms and each term must have an exponent that is 0 or a whole number. This means that \begin{align*}3x^2+2x+1\end{align*}
- A monomial is a polynomial with just one term. Examples of monomials are \begin{align*}3x\end{align*}
3x , \begin{align*}2x^2\end{align*}2x2 and \begin{align*}7\end{align*}7 . - A binomial is a polynomial with two terms. Examples of binomials are \begin{align*}2x+1\end{align*}
2x+1 , \begin{align*}3x^2-5x\end{align*}3x2−5x and \begin{align*}x-5\end{align*}x−5 . - A trinomial is a polynomial with three terms. An example of a trinomial is \begin{align*}2x^2+3x-4\end{align*}
2x2+3x−4 .
To add and subtract polynomials you will go through two steps.
- Use the distributive property to remove parentheses. Remember that when there is no number in front of the parentheses, it is like there is a 1 in front of the parentheses. Pay attention to whether or not the sign in front of the parentheses is \begin{align*}+\end{align*}
+ or \begin{align*}-\end{align*}− , because this will tell you if the number you need to distribute is \begin{align*}+1\end{align*}+1 or \begin{align*}-1\end{align*}−1 . - Combine similar terms. This means, combine the \begin{align*}x^2\end{align*}
x2 terms with the \begin{align*}x^2\end{align*}x2 terms, the \begin{align*}x\end{align*}x terms with the \begin{align*}x\end{align*}x terms, etc.
Example A
Find the sum: \begin{align*}(3x^2+2x-7)+(5x^2-3x+3)\end{align*}
Solution: First you want to remove the parentheses. Because this is an addition problem, it is like there is a \begin{align*}+1\end{align*}
\begin{align*}1(3x^2+2x-7)+1(5x^2-3x+3)=3x^2+2x-7+5x^2-3x+3\end{align*}
Next, combine the similar terms. Sometimes it can help to first reorder the expression to put the similar terms next to one another. Remember to keep the signs with the correct terms. For example, in this problem the 7 is negative and the 3x is negative.
\begin{align*}3x^2+2x-7+5x^2-3x+3&=3x^2+5x^2+2x-3x-7+3\\ &=8x^2-x-4\end{align*}
This is your final answer.
Example B
Find the difference: \begin{align*}(5x^2+8x+6)-(4x^2+5x+4)\end{align*}.
Solution: First you want to remove the parentheses. Because this is a subtraction problem, it is like there is a \begin{align*}-1\end{align*} in front of the second set of parentheses. When you distribute a \begin{align*}-1\end{align*}, each term inside that set of parentheses will change its sign.
\begin{align*}1(5x^2+8x+6)-1(4x^2+5x+4)=5x^2+8x+6-4x^2-5x-4\end{align*}
Next, combine the similar terms. Remember to keep the signs with the correct terms.
\begin{align*}5x^2+8x+6-4x^2-5x-4&=5x^2-4x^2+8x-5x+6-4\\ &=x^2+3x+2\end{align*}
This is your final answer.
Example C
Find the difference: \begin{align*}(3x^3+6x^2-7x+5)-(4x^2+3x-8)\end{align*}
Solution: First you want to remove the parentheses. Because this is a subtraction problem, it is like there is a \begin{align*}-1\end{align*} in front of the second set of parentheses. When you distribute a \begin{align*}-1\end{align*}, each term inside that set of parentheses will change its sign..
\begin{align*}1(3x^3+6x^2-7x+5)-1(4x^2+3x-8)=3x^3+6x^2-7x+5-4x^2-3x+8\end{align*}
Next, combine the similar terms. Remember to keep the signs with the correct terms.
\begin{align*}3x^3+6x^2-7x+5-4x^2-3x+8&=3x^3+6x^2-4x^2-7x-3x+5+8\\ &=3x^3+2x^2-10x+13\end{align*}
This is your final answer.
Concept Problem Revisited
Remember that the area of a rectangle is length times width.
\begin{align*}Area &= l \times w \\ Area &= (1.5x + 2) x \\ Area &= 1.5x^2 + 2x\end{align*}
Vocabulary
- Binomial
- A binomial has two terms that are added or subtracted from each other. Each of the terms of a binomial is a variable \begin{align*}(x)\end{align*}, a product of a number and a variable \begin{align*}(4x)\end{align*}, or the product of multiple variables with or without a number \begin{align*}(4x^2y + 3)\end{align*}. One of the terms in the binomial can be a number.
- Monomial
- A monomial can be a number or a variable (like \begin{align*}x\end{align*}) or can be the product of a number and a variable (like \begin{align*}3x\end{align*} or \begin{align*}3x^2\end{align*}). A monomial has only one term.
- Polynomial
- A polynomial, by definition, is also a monomial or the sum of a number of monomials. So \begin{align*}3x^2\end{align*} can be considered a polynomial, \begin{align*}2x+3\end{align*} can be considered a polynomial, and \begin{align*}2x^2+3x-4\end{align*} can be considered a polynomial.
- Trinomial
- A trinomial has three terms \begin{align*}(4x^2+3x-7)\end{align*}. The terms of a trinomial can be a variable \begin{align*}(x)\end{align*}, a product of a number and a variable \begin{align*}(3x)\end{align*}, or the product of multiple variables with or without a number \begin{align*}(4x^2)\end{align*}. One of the terms in the trinomial can be a number \begin{align*}(-7)\end{align*}.
- Variable
- A variable is an unknown quantity in a mathematical expression. It is represented by a letter. It is often referred to as the literal coefficient.
Guided Practice
1. Find the sum: \begin{align*}(2x^2+4x+3) + (x^2-3x-2)\end{align*}.
2. Find the difference: \begin{align*}(5x^2-9x+7) - (3x^2-5x+6)\end{align*}.
3. Find the sum: \begin{align*}(8x^3+5x^2-4x+2) + (4x^3+7x-5)\end{align*}.
Answers:
1. \begin{align*}(2x^2+4x+3) + (x^2-3x-2)=2x^2+4x+3+x^2-3x-2=3x^2+x+1\end{align*}
2. \begin{align*}(5x^2-9x+7) - (3x^2-5x+6)=5x^2-9x+7-3x^2+5x-6=2x^2-4x+1\end{align*}
3. \begin{align*}(8x^3+5x^2-4x+2) + (4x^3+7x-5)=8x^3+5x^2-4x+2+4x^3+7x-5=12x^3+5x^2+3x-3\end{align*}
Practice
For each problem, find the sum or difference.
- \begin{align*}(x^2+4x+5) + (2x^2+3x+7)\end{align*}
- \begin{align*}(2r^2+6r+7) - (3r^2+5r+8)\end{align*}
- \begin{align*}(3t^2-2t+4) + (2t^2+5t-3)\end{align*}
- \begin{align*}(4s^2-2s-3) - (5s^2+7s-6)\end{align*}
- \begin{align*}(5y^2+7y-3) + (-2y^2-5y+6)\end{align*}
- \begin{align*}(6x^2+36x+13) - (4x^2+13x+33)\end{align*}
- \begin{align*}(12a^2+13a+7) + (9a^2+15a+8)\end{align*}
- \begin{align*}(9y^2-17y-12) + (5y^2+12y+4)\end{align*}
- \begin{align*}(11b^2+7b-12) - (15b^2-19b-21)\end{align*}
- \begin{align*}(25x^2+17x-23) - (-14x^3-14x-11)\end{align*}
- \begin{align*}(-3y^2+10y-5) - (5y^2+5y+8)\end{align*}
- \begin{align*}(-7x^2-5x+11) + (5x^2+4x-9)\end{align*}
- \begin{align*}(9a^3-2a^2+7) + (3a^2+8a-4)\end{align*}
- \begin{align*}(3x^2-2x+4) - (x^2+x-6)\end{align*}
- \begin{align*}(4s^3+4s^2-5s-2) - (-2s^2-5s+6)\end{align*}
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Term | Definition |
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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, . |
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. |
Variable | A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n. |
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Here you'll learn how to add and subtract polynomials.