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# Adding Subtracting and Multiplying Polynomials

The word polynomial is used to describe a specific type of expression. Although polynomials contain multiple terms, they behave similar to numbers and can be manipulated as such.
Concept

## Polynomial

A polynomial is an algebraic expression that is the sum of multiple monomials, or terms. Consider the following example polynomial, written in standard form. $\begin{gathered} 4x^5 + x^3 - 9x - 84 \end{gathered}$

The largest exponent indicates the polynomial's degree, which in this case is $5.$ Polynomials with two or three terms are called binomials and trinomials, respectively. When there are more than three terms, the name polynomial is used more generally.
Rule

Just like numbers, polynomials are closed under addition and subtraction. This means, polynomials can be added and subtracted and the result is another polynomial. These operations are performed by combining like terms. Terms with the same variable and exponent are combined by adding or subtracting their coefficients.

### Rule

Find the sum of the following polynomials. $\text{-} 2x^2 + x + 3 \quad \text{and} \quad 2x^2 + 4x - 10$

To add these polynomials, like terms will be combined. It can be helpful to rearrange terms so that like terms are next to each other.
$\left( \text{-} 2x^2 + x + 3\right) + \left( 2x^2 + 4x - 10 \right)$
$\text{-} 2x^2 + x + 3 + 2x^2 + 4x - 10$
$\left(\text{-} 2x^2 + 2x^2 \right) + (x + 4x) + (3 - 10)$
$0x^2 + 5x - 7$
$5x - 7$
Thus, the sum is $5x-7.$ Notice that the given polynomials have degree $2.$ However, the new polynomial has degree $1.$ This is because the $x^2$-terms canceled each other out. Otherwise, the new polynomial would also have degree $2.$
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Exercise

Calculate the difference between the polynomials. $3x^2-4 \quad \text{and} \quad x^2-4x+2$

Show Solution
Solution
Subtracting polynomials is done by combining like terms. When we calculate the difference, it is important to distribute the negative sign to all terms in the second polynomial.
$\left(3x^2-4\right)-\left( x^2-4x+2 \right)$
$3x^2-4-x^2+4x-2$
$3x^2-x^2+4x-4-2$
$2x^2+4x-6$
Thus, the difference is $2x^2+4x-6.$
Rule

## Multiplying Polynomials

Polynomials are multiplied by using the Distributive Property. For example, for the polynomial product $\begin{gathered} (a+b)(c+d+e), \end{gathered}$ it is possible to distribute $(c+d+e)$ to every term of $(a+b).$ Then, by using the Distributive Property once more, the product can be expressed explicitly. $\begin{gathered} (a+b)(c+d+e)\\ \Downarrow\\ a(c+d+e) +b(c+d+e)\\[0.25em] \Downarrow\\[0.25em] ac+ ad +ae +bc+bd +be \end{gathered}$ When multiplying two polynomials, each term of the first polynomial multiplies with each term of the second one. This have some important consequences:

• The product of a polynomial with $n$ terms and a polynomial with $m$ terms results in $n\times m$ products. For example, the product of two binomials gives $2\times 2 = 4$ products as result.
• Since each product of terms results in a new monomial, the result of a polynomial multiplication is a sum of monomials, which is, by definition, another polynomial. Therefore, polynomials are closed under multiplication.
• When two polynomials are multiplied, the product is a new polynomial with a degree that is the sum of the degrees of the factor polynomials. This follows from the Product of Powers Property when multiplying the highest degree monomials of the factor polynomials.

### Rule

Multiplication

Find the product of the polynomials $x^3 + x^2+5$ and $x^4 + 2x+1$ by distributing $(x^4 + 2x+1)$ to each term in $x^3 + x^2+5.$ Then, determine the degree of the resulting polynomial. \begin{aligned} (x^3 \! + \! x^2 \! + \! 5)\!(x^4 \! + \! 2x \! + \! 1) =& \ \ \ \! x^3(x^4 \! + \! 2x\! + \!1)\\ &\! \! \! \! +x^2(x^4 \! + \! 2x\! + \!1)\\ &\! \! \! \! + \ 5 \ (x^4 \! + \! 2x\! + \!1)\\[1.2em] (x^3 \! + \! x^2 \! + \! 5)\!(x^4 \! + \! 2x \! + \! 1) =& \quad \ \ x^7 \! + \! 2x^4\! + \!x^3\\ &\! + \ \ x^6 \! + \! 2x^3\! + \! x^2\\ &\! +5x^4 \! + \! 10x\! + \! 5 \end{aligned} It can be seen that since both polynomials have $3$ terms, multiplying them results in $3 \times 3 = 9$ products. Nevertheless, it can be simplified by combining like terms. \begin{aligned} (x^3 \! + \! x^2 \! + \! 5)\!(x^4 \! + \! 2x \! + \! 1) =&\! \! \quad \ \ x^7 \! + \! {\color{#0000FF}{2x^4}}\! + \!{\color{#009600}{x^3}}\\ &\! \! \! + \ \ x^6 \! + \! {\color{#009600}{2x^3}}\! + \! x^2\\ &\! \! \! +{\color{#0000FF}{5x^4}} \! + \! 10x\! + \! 5\\[1.2em] (x^3 \! + \! x^2 \! + \! 5)\!(x^4 \! + \! 2x \! + \! 1) =&\! \! \quad \ \ x^7 \! + \! {\color{#0000FF}{7x^4}}\! + \!{\color{#009600}{3x^3}}\\ &\! \! \! + \ \ x^6 \! + \ x^2 \! + \! 10x\\ &\! \! \! + \ \ 5 \end{aligned} It can also be noted that the polynomial $x^{\color{#FF0000}{3}} + x^2+5$ its of degree ${\color{#FF0000}{3}}$ and the polynomial $x^{\color{#FF0000}{4}} + 2x+1$ its of degree ${\color{#FF0000}{4}}.$ When written in standard form, their product is $\begin{gathered} x^{\color{#FF0000}{7}}+ x^6 + 7x^4 + 3x^3 + x^2 + 10x+5. \end{gathered}$ This is a polynomial of degree ${\color{#FF0000}{3}}+{\color{#FF0000}{4}} = {\color{#FF0000}{7}}.$

In order to make the multiplication process systematic there are different methods that can be used, such as the FOIL Method and using tables of products. Ultimately, all of these methods are based on the Distributive Property. An example of a table of products is shown below.
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Exercise

Find a polynomial that represents the area of the rectangle.

Show Solution
Solution
The area of a rectangle is calculated using the relationship $A=\ell w.$ Here, the length is represented by the binomial $2x+2$ and the width by another binomial, $3x-5.$ We find the polynomial representing the area by multiplying the length and the width.
$A=\ell w$
$A=(2x+2)(3x-5)$
$A=6x^2-10x+6x-10$
$A=6x^2-4x-10$
Thus, the area can be represented by the polynomial $A=6x^2-4x-10.$