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. What is a polynomial in one variable? Explain with an example.
A polynomial in one variable is an algebraic expression where the variable has only non-negative integer exponents. It is made up of terms, where each term consists of a coefficient (a real number) and a variable raised to a whole number power. For example, the expression 3x² + 7x - 5 is a polynomial in one variable, 'x'. Here, the exponents (2 and 1) are whole numbers, and the coefficients (3, 7, -5) are real numbers.
2. Why are expressions like 5√t + 2 or x + 3/x not considered polynomials?
These expressions are not polynomials because they violate the fundamental rule for a variable's exponents. The power of the variable in every term must be a non-negative integer (0, 1, 2, ...).
- In the expression 5√t + 2, the term 5√t can be written as 5t1/2. The exponent is 1/2, which is a fraction, not a whole number.
- In the expression x + 3/x, the term 3/x can be written as 3x-1. The exponent is -1, which is negative.
Therefore, neither expression qualifies as a polynomial.
3. How are polynomials classified based on their degree and number of terms?
Polynomials can be classified in two primary ways:
- Based on Degree (the highest exponent of the variable):
- Linear Polynomial: Degree 1 (e.g., 5x + 1)
- Quadratic Polynomial: Degree 2 (e.g., x² - 3x + 2)
- Cubic Polynomial: Degree 3 (e.g., 4x³ - 9)
- Based on the Number of Terms:
- Monomial: One term (e.g., 7x²)
- Binomial: Two terms (e.g., 2x - 5)
- Trinomial: Three terms (e.g., 8x² + 3x - 1)
4. What is the 'degree' of a polynomial and why is it an important concept?
The degree of a polynomial is the value of the highest exponent of the variable in any of its terms. For example, in the polynomial 6x⁵ - 2x³ + 9, the degree is 5. The degree is important because it tells you fundamental information about the polynomial, such as the maximum number of 'zeroes' (or roots) it can have and the general shape of its graph. A polynomial of degree 'n' can have at most 'n' real roots.
5. Can the coefficient of a term in a polynomial be a fraction or a radical?
Yes. The condition of having non-negative integer powers applies only to the variable, not its coefficient. The coefficients can be any real number, including fractions, decimals, or irrational numbers like radicals. For instance, (1/3)x² + √5x - 8 is a perfectly valid quadratic polynomial because the exponents of x (2 and 1) are whole numbers, even though the coefficients are a fraction and a radical.
6. What is the main difference between a 'polynomial' and a 'polynomial equation'?
The main difference lies in their structure and purpose. A polynomial is simply an algebraic expression, like x² - 9. Its purpose is to be simplified or evaluated for a given value of x. In contrast, a polynomial equation is a statement that sets a polynomial equal to another expression, usually zero, such as x² - 9 = 0. The purpose of an equation is to be solved to find the values of the variable (the roots) that make the statement true.
7. How do you find the 'zero' of a polynomial? Explain with a simple example.
A 'zero' of a polynomial is any value of the variable that makes the entire polynomial equal to zero. To find the zero(es), you set the polynomial expression, P(x), equal to 0 and solve for x. For example, to find the zero of the linear polynomial P(x) = 2x - 8:
- Set the polynomial to zero: 2x - 8 = 0
- Solve for x: 2x = 8
- Result: x = 4
Therefore, 4 is the zero of the polynomial P(x) = 2x - 8.
8. What is the Remainder Theorem and how does it simplify finding remainders?
The Remainder Theorem states that if a polynomial P(x) is divided by a linear polynomial of the form (x - a), the remainder of that division will be equal to the value of P(a). This provides a shortcut to find the remainder without performing the entire long division process. For example, to find the remainder when P(x) = x² + 3x - 5 is divided by (x - 2), you simply calculate P(2): P(2) = (2)² + 3(2) - 5 = 4 + 6 - 5 = 5. The remainder is 5.
9. How is the Factor Theorem related to the Remainder Theorem?
The Factor Theorem is a direct consequence of the Remainder Theorem. It states that a linear expression (x - a) is a factor of a polynomial P(x) if and only if P(a) = 0. Essentially, if substituting 'a' into the polynomial gives a result of zero, it means the remainder upon division by (x - a) is zero. When the remainder is zero, the division is exact, which by definition means (x - a) is a factor of P(x).
