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Factorization of Polynomials

The document provides an introduction to polynomials, defining them as algebraic expressions with variables and coefficients, and categorizing them by degree and number of terms. It explains the Remainder Theorem and Factor Theorem, which are methods for finding remainders and factors of polynomials. Additionally, it outlines the process of long division for polynomials that cannot be easily factored.

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pattnaiksubhanandan3
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7 views4 pages

Factorization of Polynomials

The document provides an introduction to polynomials, defining them as algebraic expressions with variables and coefficients, and categorizing them by degree and number of terms. It explains the Remainder Theorem and Factor Theorem, which are methods for finding remainders and factors of polynomials. Additionally, it outlines the process of long division for polynomials that cannot be easily factored.

Uploaded by

pattnaiksubhanandan3
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Page 1 - Introduction to Polynomials

What is a Polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients, involving only

addition, subtraction, multiplication, and non-negative integer exponents of variables.

Examples of Polynomials:

- x² + 5x + 6

- 2a³ - 4a² + a - 7

- 7y - 3

Examples that are NOT Polynomials:

- 1 divided by x

- Square root of x

- x raised to negative 2

Key Features:

- Degree: Highest power of the variable.

- Coefficient: Number multiplied by the variable.

- Constant Term: A term without a variable.

Important Points:

- Powers must be non-negative integers.

- Variables should not be in the denominator or under roots.

Page 2 - Types of Polynomials


Classification by Degree:

- Constant Polynomial: Degree 0 (example: 5)

- Linear Polynomial: Degree 1 (example: x + 2)

- Quadratic Polynomial: Degree 2 (example: x² + 3x + 2)

- Cubic Polynomial: Degree 3 (example: x³ - x + 1)

Classification by Number of Terms:

- Monomial: 1 term

- Binomial: 2 terms

- Trinomial: 3 terms

Examples:

- 7x³ + x² is a cubic binomial.

- 3 is a constant monomial.

Important Points:

- Degree affects the number of solutions.

- Number of terms affects factorization methods.

Page 3 - Remainder Theorem

The Remainder Theorem states that if a polynomial f(x) is divided by (x - a), the remainder is f(a).

Instead of dividing manually, substitute a into f(x) to find remainder easily.

Example 1:

Find remainder when f(x) = x³ + 2x² - 5x + 1 is divided by (x - 2).


Solution:

f(2) = (2)³ + 2(2)² - 5(2) + 1

= 8 + 8 - 10 + 1

=7

Thus, the remainder is 7.

Important Points:

- Always substitute opposite sign.

- If remainder is 0, divisor is a factor.

Page 4 - Factor Theorem

The Factor Theorem is a special case of the Remainder Theorem.

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

Example 1:

Check if (x - 1) is a factor of f(x) = x² - 3x + 2.

Solution:

Substitute x = 1:

f(1) = (1)² - 3(1) + 2 = 1 - 3 + 2 = 0

Since f(1) = 0, (x - 1) is a factor.

Important Points:

- Factor Theorem is useful for factorizing higher degree polynomials.

Page 5 - Long Division of Polynomials


When polynomials cannot be factored easily, use long division.

Steps:

- Arrange terms in descending order of powers.

- Divide first term of dividend by first term of divisor.

- Multiply and subtract.

- Repeat until remainder is zero or lower degree than divisor.

Example:

Divide f(x) = x³ - 2x² + 4x - 8 by (x - 2).

Important Points:

- Always align terms properly.

- Division stops when degree of remainder is less than divisor.

... (content continues for pages 6 to 15)

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