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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 \(3\).
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 \(\frac{-b}{a}\) \(=\) \(\frac{-\left(\text{Constant term}\right)}{\text{Coefficient of }x}\).
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