Definition Degree and Solved Examples of Polynomials 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 Concepts and Standard Form
1. What is a polynomial in one variable?
A polynomial in one variable is an algebraic expression made up of terms involving powers of a single variable with non-negative integer exponents. It has the general form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where:
- x is the variable
- a₀, a₁, ..., aₙ are constants (coefficients)
- n is a non-negative integer
Example: 3x² + 2x − 5 is a polynomial in one variable x.
2. What is the degree of a polynomial in one variable?
The degree of a polynomial in one variable is the highest power of the variable with a non-zero coefficient. For example:
- In 5x⁴ − 2x² + 7, the degree is 4
- In −3x + 9, the degree is 1
- A non-zero constant like 6 has degree 0
The degree helps classify polynomials as linear, quadratic, cubic, etc.
3. What are the types of polynomials based on degree?
Polynomials in one variable are classified by degree as linear, quadratic, cubic, and higher. Common types include:
- Linear polynomial: degree 1 (e.g., 2x + 3)
- Quadratic polynomial: degree 2 (e.g., x² − 4x + 1)
- Cubic polynomial: degree 3 (e.g., x³ + 2x)
- Quartic polynomial: degree 4
This classification is important when solving polynomial equations or studying graphs of polynomials.
4. How do you find the value of a polynomial for a given value of x?
To find the value of a polynomial, substitute the given value of x and simplify the expression. For example, evaluate P(x) = 2x² − 3x + 1 at x = 2:
- P(2) = 2(2²) − 3(2) + 1
- = 2(4) − 6 + 1
- = 8 − 6 + 1
- = 3
This process is called evaluating a polynomial.
5. What is a zero of a polynomial in one variable?
A zero (root) of a polynomial is a value of x that makes the polynomial equal to zero. In other words, if P(a) = 0, then a is a zero of P(x).
Example: For P(x) = x − 4, substitute x = 4:
- P(4) = 4 − 4 = 0
So, 4 is a zero of the polynomial.
6. How do you find the zeros of a quadratic polynomial?
The zeros of a quadratic polynomial ax² + bx + c are found using the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. Steps:
- Identify a, b, and c
- Compute the discriminant D = b² − 4ac
- Substitute into the formula
Example: For x² − 5x + 6 = 0:
- D = 25 − 24 = 1
- x = (5 ± 1)/2
- Roots are 3 and 2
7. What is the remainder theorem for polynomials?
The Remainder Theorem states that when a polynomial P(x) is divided by (x − a), the remainder is P(a). Instead of long division, simply substitute x = a.
Example: For P(x) = x² − 3x + 2, find the remainder when divided by (x − 1):
- P(1) = 1 − 3 + 2 = 0
So, the remainder is 0, meaning (x − 1) is a factor.
8. What is the factor theorem in polynomials?
The Factor Theorem states that (x − a) is a factor of a polynomial P(x) if and only if P(a) = 0. It is a direct extension of the Remainder Theorem.
Example: If P(x) = x² − 4, then:
- P(2) = 4 − 4 = 0
So, (x − 2) is a factor of the polynomial.
9. How do you add and subtract polynomials in one variable?
To add or subtract polynomials, combine like terms with the same powers of the variable. Steps:
- Write polynomials in standard form
- Align like terms
- Add or subtract coefficients
Example: (2x² + 3x − 1) + (x² − 4x + 5)
- = (2x² + x²) + (3x − 4x) + (−1 + 5)
- = 3x² − x + 4
10. What is the standard form of a polynomial in one variable?
The standard form of a polynomial in one variable is when terms are arranged in descending order of powers of the variable. The general structure is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀.
Example: The expression 3 + 2x³ − x should be written in standard form as 2x³ − x + 3.
Writing polynomials in standard form makes it easier to identify the degree and leading coefficient.
