Introduction
A polynomial is an algebraic expression where the powers of the variables are non-negative integers. Imagine organizing items in boxes; each box represents a term, and the total collection is your polynomial.
Explanation Step by Step
To work with polynomials, first identify the terms. Constants, variables, and exponents combine to form these expressions. We can perform arithmetic operations just like we do with regular numbers, but we must only combine "like terms" (terms with the same variable and power).
Sub-topics
Polynomial, their operations and factors of polynomial
Operations include addition (grouping same powers), subtraction, and multiplication (using the distributive property). Factoring is the process of breaking down a polynomial into a product of simpler polynomials.
Examples
Example 1 (Real-life)
You buy n pens at 10 rupees each and pay 20 rupees for a notebook. Express the total cost.
Tricky Example
Is x2 + 3√x + 4 a polynomial?
Addition of Polynomials
Only combine like terms. Like terms have the same variable and the same exponent.
Examples
Example 1 (Real-life)
If box A has 5x pens and box B has 2x pens, total pens are 7x.
Subtraction of Polynomials
Distribute the negative sign to every term of the second polynomial.
Examples
Tricky Example
Subtract (x - 5) from (2x + 1).
Multiplication of Polynomials
Multiply the coefficients and add the exponents of the variables.
Examples
Example 1
Multiply 4x2 * 3x3.
Division of Polynomials
Divide the numerical coefficients and subtract the exponents.
Examples
Tricky Example
Divide (10x4 - 5x2) by 5x2.
Tricks and Shortcuts
- Use the FOIL method (First, Outer, Inner, Last) for multiplying two binomials.
- To find the degree of a polynomial, just look for the highest exponent.
Common Mistakes
- Adding terms with different powers, like x2 + x = x3 (This is wrong!).
- Forgetting to distribute a negative sign to all terms inside a bracket during subtraction.
Practice Questions
Easy Questions
- Simplify: 4y + 3y - 2y.
- State the degree of 5x3 + 2x2 + 1.
- (Tricky) Is a constant number like "7" a polynomial?
- Add: (3x + 4) + (2x + 5).
- Multiply: 5 * 4x2.
- (Tricky) Is 2x + 3y equal to 5xy? Explain.
Medium Questions
- Multiply (x - 5)(x + 5).
- Subtract (2x2 + 3x) from (5x2 + 7x).
- A square garden has a side of (x + 4) meters. Find its area. (Real-life)
- Subtract: (10x - 2) - (4x + 1).
- (Real-life) If a square has a side of 3x cm, find its area.
- Multiply: (x + 1)(x + 2).
Hard Questions
- Factorize x2 + 7x + 12.
- (Tricky Real-life) A shopkeeper sells x items for (x-3) rupees each. If his total revenue is x2 - 3x, find the revenue if he sells 10 items.
- Divide x2 + 5x + 6 by (x + 2).
- Divide: (6x3 + 9x2) / 3x.
- (Tricky) Simplify: (x+y)2 - (x2+y2).
- (Real-life) A rectangular park has area x2+7x+12. If the length is x+4, find the width.
Revision Summary
Polynomials consist of variables and coefficients. Key operations are addition, subtraction, and multiplication. Factoring helps in solving equations by finding the "building blocks" of an expression.
Add/Subtract like terms, Add exponents for multiplication, and Subtract exponents for division.