Introduction to Polynomial in One Variable
The word "polynomial" is derived from the terms "poly", which means "many", and "nomen" which means "name". Variables and coefficients make up a polynomial expression. Expressions that contain one or more terms with a non-zero coefficient are known as polynomials. There may be several terms in a polynomial. Each expression contained in the polynomial is known as a term. The concept of polynomials in one variable, terms connected to polynomials, and the categorisation of polynomials in one variable based on the degree are all covered in this article, along with numerous examples which have been solved.
What are Polynomials in One Variable?
The two main categories of symbols we discover in algebra are constants and variables. A symbol with a set numerical value is referred to as a constant, such as \[10,112, - 7,\dfrac{5}{7}\], while a symbol whose value changes depending on the circumstances is referred to as a variable, such as \[x,y,z,{x^2}\].
While $x$ is a variable and $3$ is a constant in $3x$, but $3x$ as a whole is a variable because its value will change as $x$ value does, and vice versa. In a similar way, $3$ is a constant, and $x$ is a variable, but $x + 3$, $x - 3$, and $x \div 3$ are all variables. As a result, we may say that any time a constant and a variable are combined, the result is always a variable.
Related Terms for Polynomials
The following is a list of the several terms associated with polynomials:
Terms: A term can be either a variable, a constant (number), or a combination of both. And based on terms, we can classify the polynomials as a monomial, binomial and trinomial.
Coefficient: A coefficient is a number or variable multiplied by another variable in the expression.
Variable: A variable is a symbol that stands in for an expression's unknown value.
Constant: In an equation, a constant is an integer whose value never changes.
Think about the example $5x + 2$. Here, $x$ is the variable, $5$ is the coefficient of x, $2$ is the constant, and $5x$ and $2$ are the terms.
Classification of Polynomials
The degree of a polynomial can be used to categorise the polynomials in a single variable. The largest power of a polynomial's variable is referred to as the polynomial's degree. A polynomial can be divided into four categories, according to its degree in a single variable:
Zero or a constant polynomial: A polynomial with degree zero is referred to as a zero or constant polynomial. Only constant terms and no variables are present in such polynomials. The zero polynomial $2$, which may also be written as $2{x^0}$, is an illustrative example.
Linear polynomial: A linear polynomial is a polynomial with a degree $1$. The maximum number of terms in a linear polynomial with one variable is two. Example: \[x + 4,y + 7\] .
Quadratic polynomial: A quadratic polynomial is a polynomial with a degree of $2$ as its highest degree. One-variable quadratic polynomials only have two solutions. Example: \[{x^2} + 5x + 10\] , \[{y^2} - 2y + 6\] .
Cubic polynomial: A polynomial is referred to as a cubic polynomial if the maximum exponent of any variable in the polynomial is $3$, or if the degree of the polynomial is $3$. The number of solutions to a cubic polynomial in one variable is three. For Example: \[10{m^3} - 5m\] , \[125{y^3} - 1\] .
Possible Solutions of Polynomial in One Variable
Finding any polynomial's degree is the first step in solving it. The biggest exponent of a polynomial with a single variable is its degree, as was previously discussed. Thus,
There is only one possible solution for linear polynomials with one variable.
There are only two possible solutions for quadratic polynomials with one variable.
There are only three possible solutions for cube polynomials in one variable.
Factorization of Polynomial in One Variable
Identity-Based Factorization
Here, we will use the algebraic identities for factorization. The following are the three identities:
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
$(a - b)(a + b) = {a^2} - {b^2}$
\[{(a - b)^2} = {a^2} - 2ab + {b^2}\]
Factorization via Middle Term Splitting
The quadratic equation of the form \[a{x^2} + bx + c\] , where $a$ , $b$ and $c$ are constants and $x \ne 0$, can be solved by splitting the middle term.
The rule to factorize \[a{x^2} + bx + c\], where $a,b,c$ are real numbers, is to split the coefficient of $x$ i.e., $b$ into two real numbers so that their algebraic sum is $b$ and their product is $ac$ . Then use the grouping approach to factorize.
The following rule can save a lot of time even though factoring a polynomial in one variable is not always feasible.
Calculate ${b^2} - 4ac$ for the equation \[a{x^2} + bx + c\] . In that case, the provided expression will factorise if it is a perfect square. Otherwise, no.
Examples of Polynomials in One Variable
Q.1. Factorise ${x^2} + 9x + 18$ .
Solution. We need to discover two real integers whose sum is $9$ and whose product is $18$ in order to factor the expression ${x^2} + 9x + 18$.
Trial and error shows that $3 + 6 = 9$ and $3 \times 6 = 18$ .
In light of this, ${x^2} + 9x + 18 = {x^2} + 3x + 6x + 18$ .
Taking common terms, we get,
${x^2} + 9x + 18 = x(x + 3) + 6(x + 3)$Solving this, we will get,
\[ = (x + 3)(x + 6)\]
The equation ${x^2} + 9x + 18$ can therefore be factored as \[(x + 3)(x + 6)\] .
Q.2. As ${x^3} + {x^7} - {x^9}$ , write the polynomial's degree.
Solution. The polynomial ${x^3} + {x^7} - {x^9}$ has a degree of $9$ because the variable $x$ in the above polynomial has the largest power of $9$ .
Summary
In this article, we have learned about the polynomial of one variable. Along with learning about cubic and quadratic polynomials, we also discovered how to calculate the polynomial degree. We now understand how to express polynomials that are linear, quadratic, cubic, and of degree $n$ in their generalized form.
Practice Questions
1. Factorise \[{x^2} + 10x + 25\] as a polynomial.
2. Determine the polynomial $22$ degree.
Answers
1) $(x + 5)$
2) $0$
FAQs on Polynomials in One Variable
1. Can a polynomial have an exponent that is negative?
Considering that a polynomial by definition has non-negative integer exponents for its variables. You might therefore create an expression with negative or fractional exponents; but, it wouldn't be referred to as a polynomial. A polynomial is not an expression with a variable that has fractional or negative exponents, divides by a variable, or is contained inside a radical. As long as the polynomial is defined over the real numbers, it is possible for a polynomial to contain negative coefficients, fractions, or even radicals.
2. How do polynomial and polynomial equations differ from one another?
An expression that consists of one or more variables, coefficients, and non-negative integer exponents of variables is called a polynomial. A mathematical statement that has two algebraic expressions with equal values and the symbol "equal to" between them is called an equation. Polynomial equations are expressions equated to zero, whereas polynomials are expressions. A polynomial can be written as the sum of a finite number of terms, where each term is the result of multiplying one or more variables by a constant coefficient and a positive integer exponent (or power).
3. What are Some Important Polynomial Theorems?
Polynomial consists of some essential properties and theorems, such as:
The first property starts with the Division theorem which means that a polynomial P(x) is divided by another polynomial and the result has turned out to be in quotient Q(x) with remainder R(x). The second theorem is Bezout's theorem where a polynomial P(x) is capable of being divided by a binomial only if P(a) = 0. The third property is the Factor theorem, where P(x) is divided by a Quotient Q(x) and provides the result R (x).
Apart from these, intermediate value theorems, Descartes' Rule of Sign, etc. are equally important factors.
