Polynomials

A polynomial is an expression made up of variables and constants, combined using only addition, subtraction, and multiplication (no division by variables).

General Form:

$$P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$$

Where:

• $a_n, a_{n-1}, \ldots, a_0$ are real coefficients
• $x$ is a variable
• $n$ is a non-negative integer and $a_n \ne 0$
• $a_nx^n$ is the leading term
• $a_0$ is the constant term

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Classification Based on Degree

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Types of Polynomials Based on Terms

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Operations on Polynomials

Addition & Subtraction

• Combine like terms

• Example:

  $$(3x^2 + 2x - 1) + (4x^2 - 3x + 5) = 7x^2 - x + 4$$

Multiplication

• Use distributive property or identity formulas

• Example:

  $$(x + 2)(x - 3) = x^2 - x - 6$$

Division

• Long division or synthetic division (for special cases)

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Identities to Remember

Algebraic Identities

$$(a + b)^2 = a^2 + 2ab + b^2 \\ (a - b)^2 = a^2 - 2ab + b^2 \\ a^2 - b^2 = (a - b)(a + b) \\ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \\ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$

Factorization Identities

$$x^2 + (a + b)x + ab = (x + a)(x + b) \\ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \\ x^3 - y^3 = (x - y)(x^2 + xy + y^2)$$

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Remainder Theorem

If a polynomial $f(x)$ is divided by $(x - a)$, the remainder is:

$$f(a)$$

Shortcut: To find the remainder when $f(x)$ is divided by $x - a$, just plug in $a$ into the polynomial.

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Factor Theorem

If $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$.

So, to test if $x - 2$ is a factor, plug in $x = 2$. If $f(2) = 0$, it’s a factor.

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Graphical Understanding

The degree of the polynomial tells us how many turns or bends the graph can have:

• Linear: straight line
• Quadratic: parabola (1 bend)
• Cubic: S-shape (up to 2 bends)

Roots of the polynomial are the points where the graph intersects the x-axis.

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Important Terms

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Common Conceptual Mistakes

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Example 1: Remainder Theorem

Find the remainder when $f(x) = x^3 - 4x + 1$ is divided by $x - 2$

$$f(2) = 8 - 8 + 1 = 1 \Rightarrow \text{Remainder} = 1$$

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Example 2: Factor Theorem

If $f(x) = x^3 - 7x + 6$, check if $x - 1$ is a factor.

$$f(1) = 1 - 7 + 6 = 0 \Rightarrow \text{Yes, it's a factor}$$

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Example 3: Polynomial Division

Divide $x^3 + 3x^2 + 3x + 1$ by $x + 1$

Using long division or observation:

$$(x + 1)^3 = x^3 + 3x^2 + 3x + 1 \Rightarrow \text{Quotient} = x^2 + 2x + 1$$

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Example 4: Word Problem

A polynomial $f(x)$ when divided by $x - 2$ gives remainder 3. What is $f(2)$?

By remainder theorem,

$$f(2) = 3$$

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Bonus Tips

• Always arrange polynomials in decreasing order of degree before any operation
• Use identities as shortcuts to save time in exams
• Factor theorems are often hidden in number-based puzzles in aptitude