Calculating with polynomials

Functions: Polynomials

Calculating with polynomials

If $f(x)$ and $g(x)$ are two polynomials and $c$ is a real number, then the following expressions also are polynomials:

$c\cdot f(x)$
$f(x)+g(x)$
$f(x)-g(x)$
$f(x)\cdot g(x)$

Rules of calculation for polynomials

Multiplying a polynomial by a constant is equivalent to multiplying each term of the polynomial by that constant.
Addition of two polynomials in $x$ is equivalent to adding the coefficients of terms with the same power of $x$ .
Subtraction of a polynomial $g(x)$ by a polynomial $f(x)$ is the same as subtracting the coefficients of terms in $g(x)$ by the coefficients of the same power of $x$ in $f(x)$ .
Multiplication of two polynomials is obtained by multiplying each term of one polynomial by each term of the other polynomial and adding all the products.

The rules specify how we can add, subtract, and multiply polynomials. The quotient $\dfrac{f(x)}{g(x)}$ of two polynomials is not always a polynomial, but does result in a rational function. We will go into this later.

On the degree of polynomials that result from arithmetic operations

Let $f(x)$ and $g(x)$ be polynomials of degree respectively $m$ and $n$ , and let $c$ be a real number.

The degree of $c\cdot f(x)$ is the degree of $f(x)$ if $c\ne0$ .
The degree of $f(x)\cdot g(x)$ is the sum of the degrees of $f(x)$ and $g(x)$ .
If $m\gt n$ , then the degree of $f(x)+g(x)$ is equal to the degree of $f(x)$ .
If $m=n$ , then the degree of $f(x)+g(x)$ is less than or equal to the degree of $f(x)$ .

To prove the statements we write $f(x)=a_mx^m+a_{m-1}x^{m-1}+\cdots+ a_0$ and $g(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots +b_0$ as above. We assume that $a_m$ and $a_n$ are not $0$ , and that $m\ge n$ .

1. The leading coefficient of $c\cdot f(x)$ is $c\cdot a_m$ ; it occurs as a coefficient of $x^m$ . Consequently, the degree of $c\cdot f(x)$ is equal to $m$ .

2. The leading coefficient of $f(x)\cdot g(x)$ is $a_m\cdot b_n$ ; it occurs as a coefficient of $x^{m+n}$ . Therefore the degree of $f(x)\cdot g(x)$ is equal to $m+n$ .

3. The leading coefficient of $f(x)+g(x)$ is $a_m$ ; it occurs as a coefficient of $x^{m}$ . Therefore the degree of $f(x)+g(x)$ is equal to $m$ .

4. The leading coefficient of $f(x)+g(x)$ is $a_m+b_m$ , unless this number is equal to zero; it occurs as a coefficient of $x^{m}$ . Therefore the degree of $f(x)+g(x)$ is equal to $m$ or, if $a_m+b_m=0$ , smaller.

What is the product of the polynomial $f(x)=x^3+x^2-7\cdot x-7$ and the constant $2$ ?

$2\cdot f(x) =$ $2\cdot x^3+2\cdot x^2-14\cdot x-14$

In order to compute the product of $f(x)$ and $2$ we multiply the coefficient of each power of $x$ in $f(x)$ with $2$ :

$\begin{array}{rcl}2\cdot f(x)&=& 2\cdot \left(x^3+x^2-7\cdot x-7\right)\\ &=&{ -14+{ (2\cdot1)\cdot x^2 }+{ (2\cdot-7)\cdot x }+{ (2\cdot-7) }} \\ &=& 2\cdot x^3+2\cdot x^2-14\cdot x-14\end{array}$

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