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Polynomial Algebra

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A polynomial is an expression of finite length constructed from variables and constants. The exponents of the terms are whole numbers and are usually ordered according to degree either in ascending order or descending powers of x. Standard form of a polynomial given below:
P(x) = $a_{n}x^{n}$ + $a_{n-1}x^{n-1}$ + ........ + $a_{2}x^{2}$ + $a_{1}$x + $a_{0}$Where, $a_{0}, a_{1}$,....... are called coefficients. $a_{0}$ is called the leading coefficient and $a_{n}$ is the constant term. x is a variable and n must be a positive integer and $a_{n} \neq$ 0.

Polynomial algebra may include the study of the linear equations, quadratic equations and the cubic equations. By linear equation, we mean the polynomial with the degree one. By quadratic polynomial, we mean the polynomial with the degree 2 and by cubic polynomial, we mean the polynomials with the degree 3.

In polynomial algebra, we can learn different methods to solve the polynomials. These methods depend upon the type of polynomial.


Some of the common methods used to find the zeroes of the given polynomials are as follows:
  • By finding the common factors.
  • By factorization.
  • By splitting the middle terms.
  • By making the polynomial in the form of perfect squares.

Some polynomials can even be solved by converting the given polynomials into the form of the standard identities and thus comparing the terms with the identities and getting the results for the polynomials.

Polynomial Functions

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A polynomial is a function of the form f(x) = $a_{n}x^{n}$ + $a_{n-1}x^{n-1}$ + ........ + $a_{2}x^{2}$ + $a_{1}$x + $a_{0}$

Degree of a polynomial is the highest power of x in its expression.

Function f(x) = 0 is also a polynomial. But, we say that its degree is undefined. Constant polynomials, linear polynomials, quadratics, cubics and quartics are polynomials of degree 0, 1, 2, 3 and 4 respectively.

Properties of Polynomials

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Given that f(x) and g(x) are polynomials with real coefficients, then the following are true:

Division Algorithm: If polynomial f(x) is divided by polynomial D(x), then it results in quotient Q(x) with remainder R(x). Then, we may write f(x) = D(x).Q(x) + R(x).

Remainder Theorem: If a polynomial f(x) is divided by a linear polynomial (x - c) with remainder r, then f(c) = r.

Factor Theorem: f(x) divided by g(x) results in h(x) with zero as remainder if and only if g(x) is a factor of f(x).

Degree: In a polynomial, each term has a degree equal to its variable's highest exponent. For example, "$x^{3}$" is a term with degree three. A polynomial's degree is equal to the power of highest power term. For terms with multiple variables, the order is equal to the sum of the exponents. For example, the degree of polynomial "$x^{2}y^{2}+x^{2}-xy-1$" is "four". This is because, the highest exponents term is $x^{2}y^{2}$, whose power adds to four.

Degree of a constant polynomial will always be zero.

Property of Zeros: Identifying the number of zeros of a polynomial, tells us how many solutions exist. For a polynomial function, if f(a) = 0, then "a" is a solution to f(x) and (x - a) is a factor of f(x). "a" is also known as a zero of the polynomial. A polynomial has number of zeros equal to its degree. For example, a polynomial of degree 3 must have 3 zero and hence it has 3 solutions. Similarly, a polynomial of degree 2 has 2 zeros.

Types of Polynomials

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Given below are the different types of polynomials:

Monomial: Polynomials which have only one term are known as monomials.
Examples: 5x, 3, -225xyz

Binomial: Polynomials which have only two terms are termed as binomials.
Examples: 5x + 7, -255xy - 22z, 75x + 5y

Trinomial: Polynomials having three terms are named as trinomials.
Examples: $y^{2}$ + 5y + 7, 3 - 2x + $5x^{3}$

Operations with Polynomials

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Operations with polynomials are explained below:

Addition of Polynomials
To add polynomials, combine like terms and add the numerical coefficients of like terms.

Let us simplify: (5x + 5y) + (3x - 4y)
First, we group the like terms and then simplify.
(5x + 5y) + (3x - 4y)
= 5x + 5y + 3x - 4y
= 5x + 3x + 5y - 4y
= 8x + y

Subtraction of Polynomials
To subtract polynomials, change the algebraic signs of the second polynomial which is to be subtracted and add it to the other.

Let us subtract l(x) = $x^{7}$ - 10$x^{6}$ + 40$x^{5}$ - 96$x^{4}$ + 176$x^{3}$ - 224$x^{2}$ + 128x from m(x) = -120$x^{5}$ -146$x^{4}$ - $x^{3}$ + 27$x^{2}$ + x - 1

Difference of polynomials l(x) and m(x) is
l(x) - m(x) = $x^{7}$ - 104$x^{6}$ + 40$x^{5}$ - 96$x^{4}$ + 176$x^{3}$ - 224$x^{2}$ + 128x - ( -120$x^{5}$ -146$x^{4}$ - $x^{3}$ + 27$x^{2}$ + x - 1)
= $x^{7}$ - 104$x^{6}$ + 40$x^{5}$ - 96$x^{4}$ + 176$x^{3}$ - 224$x^{2}$ + 128x + 120$x^{5}$ + 146$x^{4}$ + $x^{3}$ - 27$x^{2}$ - x + 1
= $x^{7}$ - 104$x^{6}$ + 160$x^{5}$ + 50$x^{4}$ +177$x^{3}$ -251$x^{2}$ +127x - 1

Multiplication of Polynomials
To multiply two polynomials when each one has more than one term, multiply each term of one polynomial with each term of the other polynomial and write like terms together.

Let a(x) = -3a, b(x) = 5ab + 7ab$^{3}$ + 8a$^{2}$b. Let us find the product of a(x) and b(x).
Product of polynomials a(x) and b(x) is
a(x) . b(x) = -3a(5ab) - 3a(7ab$^{3}$) - 3a(8a$^{2}$b)
= -15a$^{2}$b - 21a$^{2}b^{3}$ -24a$^{3}$b

Division of Polynomials

  1. To divide a monomial by another monomial, divide the numerical coefficients and literal coefficient separately.
  2. To divide a polynomial by a monomial, divide each term in the polynomial by the monomial.
Let us divide 45x$^{5}y^{3}$ by 15xy

Divide the numerical coefficient and write their quotient.

$\frac{45}{15}$ = 3

Now, divide literal coefficient and write their quotient as follows:

$\frac{x^{5}y^{3}}{xy}$ = $x^{4}y^{2}$

Write the coefficients next to each other to denote their product 3$x^{4}y^{2}$

Graphing Polynomial Functions

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A relation between some variable to other variable under some given conditions to limit its behavior is called the polynomial function.

Polynomial function is in the form f(x) = y = $A_{0}x_{n}$ + $A_{1}x_{n-1}$ + .....+ $A_{n-1}$x + $A_{n}$

Given below is the table describing the graph of the polynomial:

Degree of the graph
Equation of the graph Type of the graph
0 degree polynomial f(x) = $a_{0}$
A horizontal line with y-intercept $a_{0}$
1 degree polynomial f(x) = $a_{0}$ + $a_{1}$x Oblique line with y-intercept $a_{0}$ and slope $a_{1}$
2 degree polynomial f(x) = $a_{0}$ + $a_{1}$x + $a_{2}x^{2}$ Parabola
3 degree polynomial f(x) = $a_{0}$ + $a_{1}$x + $a_{2}x^{2}$ + $a_{3}x^{3}$ Cubic curve
Polynomial with degree 2 or greater f(x) = $a_{0}$ + $a_{1}$x + $a_{2}x^{2}$ + $a_{3}x^{3}$ + ...... + $a_{n}x^{n}$ Continuous nonlinear curve

While graphing any polynomial function, it is necessary to know the degree of the polynomial as well as the end behavior. Through degree of polynomial, we get to know the shape of the function and about the number of x-intercepts. With the help of end behavior, we come to know about the behavior of the polynomial at the end point of the graph.

Consider f(x) = x$^{2}$ - 2x - 3 = 0

Graphing Polynomial Functions

The x intercept of this function is x = -1, 3
In the given expression, the leading term has the positive coefficient and even exponent. So, the function always go towards the upside of the graph.

Polynomial Examples

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Given below are some of the example problems based on polynomials.

Solved Examples

Question 1: Consider y = 2x$^{3}$ - 5x$^{2}$ + 7x + 3. Find the value of y at x = -2.
Solution:
Plug in x = - 2 in the given equation.
We get y = 2(-2)$^{3}$ - 5(-2)$^{2}$ + 7(-2) + 3
y = -16 -20 -14 +3
y = -47

Question 2: Solve 7x$^{2}$ + 4x + 2 = 0
Solution:
Step 1:
For the given equation, the coefficients of a, b and c are 7, 4 and 2 respectively.
The given problem is solved using quadratic formula and the formula is given below:

x = $\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Step 2:
Plugging in the values of a, b and c in the above formula, we get

x = $\frac{-4\pm\sqrt{4^{2}-4 \times 7 \times 2}}{2 \times 7}$

Step 3:
Simplify the expression under the square root

x = $\frac{-4\pm\sqrt{-40}}{14}$

Solving for x, we get

$x_{1}$ = $\frac{-4+\sqrt{-40}}{14}$

$x_{1}$ = - $\frac{2}{7}$ + $\frac{1}{7}$ $i\sqrt{10}$

$x_{2}$ = $\frac{-4-\sqrt{-40}}{14}$

$x_{2}$ = - $\frac{2}{7}$ - $\frac{1}{7}$ $i\sqrt{10}$

Step 4:
Therefore, the solutions for the given equation are as follows:
$x_{1}$ = - $\frac{2}{7}$ + $\frac{1}{7}$ $i\sqrt{10}$ and $x_{2}$ = - $\frac{2}{7}$ - $\frac{1}{7}$ $i\sqrt{10}$