Algebra Class 10: Polynomial Factors

Questions and Answers

What is the complete factored form of $10 + 270y^3$?

10(1 - 3y)(1 - 3y + 9y^2)

10(1 + 3y)(1 - 3y + 9y^2) (correct)

10(1 + 3y)(1 + 3y + 9y^2)

10(1 - 3y)(1 + 3y + 9y^2)

What is the other side of a rectangular garden with area $9t - 64$ square units and one side $3t - 8$?

3t - 8

3t + 8

t - 8

t + 8 (correct)

Which expression represents the factored form of $27a^3b - 9a^3b$?

9a^3b(3 - 1)

9a^3b(3(3 - 1)) (correct)

9a^3b(3^2 - 1)

9a^3b(3^2 + 3)

What is the correct factored expression of $x^2 - 121y$?

$(x + 11)(x - 11)$ Signup and view all the answers

What is the factored form of $x^3 - 27$?

$(x - 3)(x^2 + 3x + 9)$ Signup and view all the answers

Which expression is a common factor in $12x - 27x$?

3 Signup and view all the answers

When completely factored, what form does $128 - 200m$ take?

$8(16 - 25m)$ Signup and view all the answers

What is the factored form of $4x^2 - 64$?

4(x^2 - 16)$ Signup and view all the answers

Which is the factored form of $8h + 27j$?

$(2h + 3j)(2h - 3j)$ Signup and view all the answers

Study Notes

Factoring Polynomials

Factoring helps to simplify expressions and solve equations by breaking them down into their components.
Common scenarios for factoring include the difference of two squares, sum and difference of cubes, and using the greatest common factor (GCF).

Difference of Two Squares

Formula: ( a^2 - b^2 = (a - b)(a + b) ).
Can be used with polynomials, e.g., ( x^2 - 81 = (x - 9)(x + 9) ).

Patterns in Factoring

Identification of patterns aids in factoring effectively:
- Example 1: ( 4x^2 - 49 = (2x - 7)(2x + 7) ).
- Example 2: ( 16x^2 - 81y^2 = (4x - 9y)(4x + 9y) ).

Common Monomial Factor

The process begins by identifying the GCF of all terms:
- Example: For ( 3x^2 - 12y^2 ), the GCF is ( 3 ), so it factors to ( 3(x^2 - 4y^2) ).

Polynomial Expressions

Certain binomials remain factorable after initial factoring:
- ( 1 - 16x^8 = (1 + 4x^4)(1 - 4x^4) ), where ( (1 - 4x^4) ) can further reduce.

Greatest Common Factor (GCF)

Important for simplifying expressions or finding factors:
- GCF of 24 and 54 is 6, while GCF of 20, 24, and 40 is 4.
- GCF of monomials can often be represented as the lowest power of common variables.

Factorization Examples

( 7x - 7 = 7(x - 1) ).
Area of a rectangle problem: Given area ( 9t - 64 ) and one side ( 3t - 8 ), the other side can be found using division.

Additional Activities

Encourage practice to factor expressions completely:
- ( 27ab - 9ab = 9ab(3 - 1) ).
- ( x - 121y = (x - 11y)(x + 11y) ) demonstrates using difference of squares.

Learning Objectives

Master patterns in factoring polynomials.
Factor completely using greatest common monomial factors.
Understand and apply techniques for the difference of squares and sum/difference of cubes.

Self-Assessment

Questions to check understanding of GCF and factoring processes.
Engage with multiple-choice questions and apply knowledge practically through examples.

Description

This quiz focuses on factoring polynomials, specifically the given problems related to identifying the correct factors from provided options. Test your understanding of polynomial expressions and enhance your algebraic skills.