The fundamental theorem of algebra is a theorem that introduces us to some specific characteristics of polynomials. It is one of the most basic but very important theorems in algebra. This theorem is the basis of modern algebra, and also, having the knowledge of this theorem is essential for higher Math education/learning, including trigonometry, calculus, and many others. So, what does this theorem say? It says that,
“Every polynomial of degree n will have n number of roots, and we may need to use complex numbers if required.”
Now, if you are not familiar with algebra, you may be confused with these new terms. So, let us first understand these terms and the theorem in a much simpler way.
What is a Polynomial?
You may have come across numbers such as 10×2+3×2+9 or 7×3+3×2+2x+5 somewhere. These are called polynomials, where 10×6+3×2+9 is a polynomial with 3 terms, 10×6, 3×2, 9 and 7×3+3×2+2x+5 is a polynomial with 4 terms, 7×3, 3×2, 2x, 5. These are a few of the innumerable polynomials which exist in the number system. The word polynomial is a combination of two Greek words ‘poly’ and ‘nomial’, where ‘poly’ means many, and ‘nomial’ means ‘terms’. There can be many terms, but they have to be fixed or defined because polynomials cannot contain infinite terms.
Therefore, polynomials can be defined as expressions which consist of constants, variables, and exponents. These expressions can perform the arithmetic operations of addition, subtraction, non-negative integer exponentiation, and multiplication, but no division on the variables.
- Constants are specific numbers, which accompany the variable. For example, the constants in 10×6+3×2+9 are 10, 3, and 9.
- Variables can be defined as the symbols for the numbers that we don’t know yet. We generally use alphabets as variables in algebraic expressions. So, the variable in 10×6+3×2+9 is x.
- Exponents are the powers on the variables, i.e., the number of times the base variable is multiplied to itself. For example, the exponents in 10×6+3×2+9 are 6, 2, and 0 for the first, second, and the third term, respectively.
Polynomials can also be represented as P(x), where the variable in the expression is x. So, P(x) = 10×6 + 3×2 + 9.
Polynomials can be divided based on the number of terms in that polynomial. Polynomials are called monomials if they contain only one term. The polynomial is called a binomial if it consists of two terms, and trinomial if it consists of three terms, etc.
What is the Degree of a Polynomial?
The degree of the polynomial is the highest exponent of a variable, with a non-zero coefficient in the polynomial. The word ‘order’ was often considered synonymous to ‘degree’. But ‘order’ is used to refer to some other properties of the polynomial as well, which may create some ambiguities. Therefore, it is better to use the term ‘degree’ to refer to the highest exponent.
We can divide the polynomial into different types based on its degree.
- A constant polynomial is a polynomial of degree 0.
- A linear polynomial is a polynomial of degree 1.
- A quadratic polynomial is a polynomial of degree 2.
- A cubic polynomial is a polynomial of degree 3.
What are the Roots of a Polynomial?
The roots of a polynomial are the values of variables, which result in the polynomial to become zero. Roots can be considered as the solutions of the equation, where if we put the value of root in all the variables, the equation will result in 0.
Polynomials can be represented as a general term: anxn+an-1xn-1+…+a1x+a0.
These roots of polynomials are therefore, often called ‘zeroes’ of the polynomial for the same reason, as when applied to the expression, the result is a zero.
If we happen to find the roots of the polynomial, we can also find all the factors of the polynomial and vice-versa. For example, if the polynomial of a degree 2 has roots r1 and r2, and the variable is x then the factors are (x – r1) and (x – r2).
The fundamental theorem of algebra as we read above, mentions that every polynomial of a degree ‘n’ will always have ‘n’ number of roots. The fundamental theorem is also sometimes stated as
“Every non-constant polynomial which contains complex coefficients and a degree which is greater than or equal to one has at least one root in the set of complex numbers.”
Real numbers are at core a subset of complex numbers. This can be seen from the fact that every real number can also be represented as a + bi, where a = the real number and b = 0.
What are Complex Numbers?
In the most simple terms, complex numbers are nothing but a combination of a real number and an imaginary number. As we know, real numbers are the number that we generally use, that are whole numbers (0, 1, 2, 3 …), rational numbers ( 0.2, 7/5, …), and irrational numbers (, 7, …). Real numbers can be positive, negative, or zero.
Now the question that arises is what is an imaginary number. Imaginary numbers can be described as the square roots of negative numbers or numbers which when multiplied with itself give a negative output. Imaginary numbers are represented by multiples of -1, which is called iota and represented by i. The square of i is -1. Some of the examples of complex numbers are 10i, -4 + 2i, and 6 – 5i. One important point that you need to remember is that as you know, i is -1, therefore, i2 is -1. Similarly, i3 is i2*i, which is -i and i4 is i2*i2 which happens to be 1.
As you can notice in the above examples, we combine natural numbers with imaginary numbers to form complex numbers. In 6 – 5i, the real number component is 6, and the imaginary component is -5.
If the real part of the complex number is 0, the number is called purely imaginary. For example, 10i, where the real number component is 0, and therefore, the complex number is called purely imaginary. Whereas, if the imaginary part of the complex number is 0, we consider the complex number as purely real. For example, 12, where the imaginary component is 0, and therefore, the complex number is called purely real.
As we read that roots of the polynomial can be real or complex, one fact that we should know here is that complex roots always occur in pairs. This means that if we have a complex number, it will be a conjugate pair.
Conjugate pairs are the pair of complex numbers where we can change the sign in the middle to get the other one. For example, if one part of the conjugate pair happens to be 5 + 6i, then the other part of the conjugate pair is 5 – 6i. Here we just changed the sign in the middle to obtain the other part. The other part of the conjugate pair is called the conjugate inverse.
Therefore, if one of the roots of a polynomial is a complex number, the other root will definitely be the conjugate inverse of that complex number. So, if you happen to find one of the roots to be a + bi, then the other root will be a – bi.
Now that you have a fair idea of what a polynomial is, the degree of a polynomial, the roots of a polynomial, and a broad idea of what complex numbers are, it will be easier for you to understand the fundamental theorem of Algebra and all other concepts related to it. The concepts you learned here will be the basis of your learning of algebra, and these concepts are essential for solving questions.
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