Comprehensive theory, key formulas, diagrams, and memory aids for Polynomials.
Polynomials are fundamental algebraic expressions that model a wide range of real-world situations, from calculating areas and volumes to determining the trajectory of objects in motion. In this chapter, we will explore polynomials in one variable, their degrees, zeroes, and important theorems like the Remainder and Factor Theorems.
An algebraic expression consisting of variables and constants, involving addition, subtraction, multiplication, and non-negative integer exponents of variables is called a polynomial.
If a polynomial contains only one variable (say, $x$), it is called a polynomial in one variable. The general form of a polynomial in one variable $x$ of degree $n$ is: $p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ Where $a_n, a_{n-1}, \dots, a_0$ are constants and $a_n \neq 0$.
Note: An expression is NOT a polynomial if the variable has a negative exponent or a fractional exponent. For instance, $x^{-2} + 3$ or $\sqrt{x} + 2$ are not polynomials.
Polynomials can be classified based on the number of terms or their degree.
The highest power of the variable in a polynomial is called its degree. 1. Linear Polynomial: A polynomial of degree 1. General form: $ax + b$ (where $a \neq 0$). Example: $2x - 5$. 2. Quadratic Polynomial: A polynomial of degree 2. General form: $ax^2 + bx + c$ (where $a \neq 0$). Example: $3x^2 - 4x + 1$. 3. Cubic Polynomial: A polynomial of degree 3. General form: $ax^3 + bx^2 + cx + d$ (where $a \neq 0$). Example: $x^3 - 3x^2 + 2x - 1$.
Consider a polynomial $p(x) = x - 2$. What happens when we substitute $x = 2$? $p(2) = 2 - 2 = 0$. Since the value of the polynomial becomes zero at $x = 2$, we say that $2$ is the zero of the polynomial $p(x)$.
A real number $k$ is said to be a zero of a polynomial $p(x)$ if $p(k) = 0$. Finding the zero of a polynomial $p(x)$ means solving the equation $p(x) = 0$.
When we divide a number by another, we get a quotient and a remainder. The same principle applies to polynomials.
Dividend = (Divisor × Quotient) + Remainder
If we divide a polynomial $p(x)$ by a linear polynomial $(x - a)$, the division process can be tedious. The Remainder Theorem gives us a direct way to find the remainder.
Statement of Remainder Theorem: Let $p(x)$ be any polynomial of degree greater than or equal to one and let $a$ be any real number. If $p(x)$ is divided by the linear polynomial $x - a$, then the remainder is $p(a)$.
Example: Find the remainder when $x^4 + x^3 - 2x^2 + x + 1$ is divided by $x - 1$. Here, $p(x) = x^4 + x^3 - 2x^2 + x + 1$ and the divisor is $x - 1$. So, $a = 1$. By Remainder Theorem, the remainder is $p(1)$. $p(1) = (1)^4 + (1)^3 - 2(1)^2 + 1 + 1 = 1 + 1 - 2 + 1 + 1 = 2$.
The Factor Theorem is a direct consequence of the Remainder Theorem. If the remainder when dividing $p(x)$ by $(x - a)$ is $0$, it means $(x - a)$ divides $p(x)$ perfectly, making it a factor.
Statement of Factor Theorem: If $p(x)$ is a polynomial of degree $n \ge 1$ and $a$ is any real number, then: 1. $(x - a)$ is a factor of $p(x)$, if $p(a) = 0$. 2. $p(a) = 0$, if $(x - a)$ is a factor of $p(x)$.
Example: Examine whether $x + 2$ is a factor of $x^3 + 3x^2 + 5x + 6$. Let $p(x) = x^3 + 3x^2 + 5x + 6$. The zero of $x + 2$ is $-2$. $p(-2) = (-2)^3 + 3(-2)^2 + 5(-2) + 6$ $p(-2) = -8 + 12 - 10 + 6 = 0$. Since $p(-2) = 0$, by the Factor Theorem, $(x + 2)$ is a factor of the given polynomial.
Factorisation is the reverse process of multiplication. It involves breaking a polynomial down into a product of its factors.
To factorise a quadratic polynomial of the form $ax^2 + bx + c$, we find two numbers $p$ and $q$ such that: $p + q = b$ (the coefficient of $x$) $p \times q = a \times c$ (product of the coefficient of $x^2$ and the constant term)
Example: Factorise $6x^2 + 17x + 5$. Here, $a = 6, b = 17, c = 5$. We need two numbers whose sum is $17$ and product is $6 \times 5 = 30$. The numbers are $15$ and $2$. So, we rewrite $17x$ as $15x + 2x$: $6x^2 + 15x + 2x + 5$ $= 3x(2x + 5) + 1(2x + 5)$ $= (3x + 1)(2x + 5)$.
For cubic polynomials like $ax^3 + bx^2 + cx + d$, we can use the Factor Theorem. First, guess one root (say $x=a$) by trial and error so that $p(a)=0$. Then divide the cubic polynomial by $(x-a)$ to get a quadratic quotient, which can then be factorised by splitting the middle term.
Algebraic identities are equations that are true for all values of the variables. They are incredibly useful for both expanding expressions and factorising polynomials.
Special Case: If $x + y + z = 0$, then $x^3 + y^3 + z^3 = 3xyz$.
Understanding polynomials allows us to manipulate abstract variables and solve complex equations. Mastery over the Remainder Theorem, Factor Theorem, and algebraic identities forms the core toolkit required to factorise and simplify higher-degree expressions.