Mathematics 8 Quarterly Examination

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Questions and Answers

What is the process of finding the factors of an expression?

Factoring (correct)

Special Product

Continuous Division

Rationalization

What is the GCF of 20, 24, and 40?

4 (correct)

What is the GCF of $xy^6$ and $x^3y$?

$xy^6$

$x^3y^6$

$x^3y$

$xy$ (correct)

What are the complete factors of $7x + 7$?

$7(x - 1)$ Signup and view all the answers

Find the complete factors of $x^2 - x^9$.

$x^2(1 - x^7)$ Signup and view all the answers

Find the complete factors of $2x^6 - 12x^4$.

$2x^4(x^2 - 6)$ Signup and view all the answers

If one factor of $4x^2 + 6$ is 2, what is the other factor?

$2x^2 + 3$ Signup and view all the answers

Which of the following is a perfect square expression?

$4x^2$ Signup and view all the answers

Which of the following is a perfect square?

$16x^2y^2$ Signup and view all the answers

Which of the following expressions has factors $(2x - y)(2x + y)$?

$4x^2 - y^2$ Signup and view all the answers

If one factor of the difference of two squares is $x + 2$, what is the other factor?

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

What is the complete factored form of $x^2 - 16$?

($x + 4)(x - 4) Signup and view all the answers

What is the complete factored form of $x^2y^2 - 1$?

($xy + 1)(xy - 1) Signup and view all the answers

Which of the following is a perfect cube?

$8x$ Signup and view all the answers

Which of the following is the complete factored form of $x^3 - 8$?

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

Using the pattern for factoring the sum of cubes, we know that factoring $8 + b^3$ gives _____.

((2 + b)(4 + 2b + b^2)) Signup and view all the answers

What is the complete factored form of $x^3y^3 + 1$?

($xy + 1)(x^2y^2 - xy - 1) Signup and view all the answers

What is the complete factored form of $27 - x^6$?

($3 - x^2)(9 - 3x^2 + x^4) Signup and view all the answers

Factor completely: $64x^3 - y^3$.

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

If one factor of $x^6 + 1000$ is $x^4 - 10x^2 + 100$, what is the other factor?

$x^2 + 100$ Signup and view all the answers

Supply the missing expression to make it true, $10 + 270y^3 = 10(____)(1-3y + 9y^2).

$1 + 3y$ Signup and view all the answers

Which of the following is a perfect square trinomial?

$8a^2 - 6a + 1$ Signup and view all the answers

Study Notes

Exam Instructions

Read each question thoroughly.
Shade the letter corresponding to the selected answer.
Do not use mobile devices during the examination.

Topics Covered

Factoring Processes
- Finding factors of an expression is known as factoring.
- Processes include special products, rationalization, and continuous division.

Greatest Common Factor (GCF)

GCF of numbers like 20, 24, and 40 is 4.
For variables, find the lowest exponent (e.g., GCF of (xy^6) and (x^3y) is (xy)).

Factoring Expressions

Factors of (7x + 7) include (7(x + 1)) or (7(x - 1)).
Quotients related to polynomial factoring are explored through expressions like (x^2 - x^9) which can be factored as (x^2(x - x^7)).

Perfect Squares and Cubes

Recognize perfect squares such as (4x^2) and (16x^2y^2).
Understand perfect cubes including (8x) and (64x^4).

Difference and Sum of Squares

The difference of two squares can be factored using the relationship (a^2 - b^2 = (a - b)(a + b)).
For example, (x^2 - 16) can be factored into ((x - 4)(x + 4)).

Factoring Techniques

Use patterns for special products (e.g., sum of cubes or difference of cubes).
For example, the expression (x^3 - 8) uses the formula ( (a^3 - b^3 = (a - b)(a^2 + ab + b^2)).

Special Factorizations

Sum of cubes can be applied to (x^3y^3 + 1) resulting in ((xy + 1)(x^2y^2 - xy + 1)).
Recognize complete factored forms, like (27 - x^6) that can be transformed as ( (3 - x^2)(9 + 3x^2 + x^4) ).

Exam Question Formats

Multiple-choice format emphasizing various mathematical concepts.
Applications of factoring principles and recognizing polynomial forms are critical to answering correctly.

Important Formulas

Difference of squares: (a^2 - b^2 = (a - b)(a + b))
Perfect square trinomial: (a^2 \pm 2ab + b^2 = (a \pm b)^2)
Sum of cubes: (a^3 + b^3 = (a + b)(a^2 - ab + b^2))

Practice Problems

Engage with problems that require the application of these factoring methods and recognition of perfect squares and cubes for effective preparation.

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Description

Test your knowledge with this Mathematics 8 quarterly exam covering essential concepts such as factoring and greatest common factors (GCF). Read each question carefully and ensure you are familiar with the topics. Good luck!