In Class IX, you were introduced to polynomials and learned about their basic concepts such as degree, zeroes, and factorisation. In this chapter, we further extend our study by exploring the relationships between zeroes and coefficients and applying polynomials to form and solve quadratic equations.
Diophantus of Alexandria, the "Father of Algebra/Polynomials," revolutionized polynomial work through his book Arithmetica, introducing symbolic notation (syncopated style) for unknowns and powers, moving algebra from words to symbols, and focusing on finding rational/integer solutions (Diophantine equations) for polynomial equations, laying groundwork for modern algebra and number theory.
Let us recall some of the basic concepts of polynomials.
Degree of the polynomial
If \(p(x)\) is a polynomial in \(x\), the highest power of \(x\) in \(p(x)\) is called the degree of the polynomial \(p(x)\).
Example:
\(p(x)\) \(=\) \(5x^3 + 2\)
The highest power of the polynomial \(p(x)\) is \(3\).
Therefore, the degree of \(p(x)\) is \(3\).
Important!
It's not defined the degree of zero polynomial. There can be any degree. can be substituted as — where '\(n\)' can be any number.
For example: \(p(x) = 0 × x^6 = 0\).
The constant polynomial is the form \(p(x) = c\), where \(c\) is the actual number. This means that it is constant for all possible values of \(x\), \(p(x) = c\).
For example: \(p(x) = 6 = 6 x^0\) [where \(x^0 = 1\)]
Note that the highest power of the '\(x\)' is zero.
Therefore, the degree of the non-zero constant polynomial is zero.
Linear polynomial
A polynomial of degree \(1\) is called a linear polynomial.
Example:
\(p(x)\) \(=\) \(2x + 3\)
Quadratic polynomial
A polynomial of degree \(2\) is called a quadratic polynomial.
Example:
\(p(x)\) \(=\) \(2x^2 + x + 3\)
Cubic polynomial
A polynomial of degree \(3\) is called a cubic polynomial.
Example:
\(p(x)\) \(=\) \(2x^3 + 3x^2 - 6x + 3\)
Value of the polynomial
If \(p(x)\) is a polynomial in \(x\), then for any real number \(k\) the value obtained by replacing \(x\) by \(k\) in \(p(x)\) is the value of the polynomial \(p(x)\) at \(x=k\).
The value of the polynomial \(p(x)\) at \(x=k\) is denoted by \(p(k)\).
Zero of the polynomial
If the value of the polynomial \(p(x)\) at \(x = k\) is zero \((p(k) = 0)\), then the real number \(k\) is called the zero of the polynomial \(p(x)\).
Zero of a linear polynomial
If \(k\) is the zero of the linear polynomial \(p(x)\) \(=\) \(ax +b\), then \(p(k)\) \(=\) \(0\).
This implies \(ak +b\) \(=\) \(0\)
So, \(k\) \(=\) \(\frac{-b}{a}\).
In general, the zero of the linear polynomial \(p(x)\) \(=\) \(ax +b\) is given by \(\frac{-b}{a}\) \(=\) \(\frac{-\left(\text{Constant term}\right)}{\text{Coefficient of }x}\).