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Questions and Answers
What are the factors of the quadratic trinomial $x^2 + 5x + 6$?
What are the factors of the quadratic trinomial $x^2 + 5x + 6$?
- (x + 1)(x + 6)
- (x + 6)(x - 1)
- (x - 2)(x + 3)
- (x + 2)(x + 3) (correct)
The expression $2x^2 - 9x + 9$ can be factored as $(x - 3)(2x - 3)$.
The expression $2x^2 - 9x + 9$ can be factored as $(x - 3)(2x - 3)$.
True (A)
What is the product of the coefficient of the square term and the constant term in the expression $2x^2 - 9x + 9$?
What is the product of the coefficient of the square term and the constant term in the expression $2x^2 - 9x + 9$?
18
The expression $p^2 - 9q^2$ can be factored as _____ and _____.
The expression $p^2 - 9q^2$ can be factored as _____ and _____.
Match the following expressions with their factored forms:
Match the following expressions with their factored forms:
Which expression is recognized as a quadratic trinomial?
Which expression is recognized as a quadratic trinomial?
The factors of $x^2 + (a + b)x + ab$ are $(x + a)$ and $(x + b)$.
The factors of $x^2 + (a + b)x + ab$ are $(x + a)$ and $(x + b)$.
What is the sum of the two numbers that factor the quadratic $x^2 + 5x + 6$?
What is the sum of the two numbers that factor the quadratic $x^2 + 5x + 6$?
What is the first step in simplifying $(2x + 3y)^3 - (2x - 3y)^3$ using the difference of cubes formula?
What is the first step in simplifying $(2x + 3y)^3 - (2x - 3y)^3$ using the difference of cubes formula?
The difference of cubes formula can be applied to simplify the expression $(x + y)^3 - (x - y)^3$.
The difference of cubes formula can be applied to simplify the expression $(x + y)^3 - (x - y)^3$.
What is the simplified form of $(x + y)^3 - (x - y)^3$?
What is the simplified form of $(x + y)^3 - (x - y)^3$?
The formula for $a^3 - b^3$ is $(a - b)(a^2 + ab + b^2)$. Thus, $(2x + 3y)^3 - (2x - 3y)^3 = [(2x + 3y) - (2x - 3y)][__________].
The formula for $a^3 - b^3$ is $(a - b)(a^2 + ab + b^2)$. Thus, $(2x + 3y)^3 - (2x - 3y)^3 = [(2x + 3y) - (2x - 3y)][__________].
What is the factored form of $2y^2 - 4y - 30$?
What is the factored form of $2y^2 - 4y - 30$?
Match the following algebraic expressions with their corresponding simplifications:
Match the following algebraic expressions with their corresponding simplifications:
Which of the following gives a rational algebraic expression?
Which of the following gives a rational algebraic expression?
The sum of cubes formula can be expressed as $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
The sum of cubes formula can be expressed as $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
Dividing algebraic expressions can yield non-zero denominators.
Dividing algebraic expressions can yield non-zero denominators.
What is the first step in factoring $x^2 + 9x + 18$?
What is the first step in factoring $x^2 + 9x + 18$?
The expression $x^3 + 27y^3$ can be factored as $(x + 3y) [x^2 - 3xy + 9y^2]$. This is based on the __________ formula.
The expression $x^3 + 27y^3$ can be factored as $(x + 3y) [x^2 - 3xy + 9y^2]$. This is based on the __________ formula.
Factor the expression $27m^3 - 216n^3$.
Factor the expression $27m^3 - 216n^3$.
Match the following expressions with their factored forms:
Match the following expressions with their factored forms:
Which of the following expressions can be factored as $(a + b)(a^2 - ab + b^2)$?
Which of the following expressions can be factored as $(a + b)(a^2 - ab + b^2)$?
The expression $x^2 - 10x + 9$ can be factored as $(x - 1)(x - 9)$.
The expression $x^2 - 10x + 9$ can be factored as $(x - 1)(x - 9)$.
When factoring $m^2 - 25m + 100$, the expression can be written as $(m - ______)^2$.
When factoring $m^2 - 25m + 100$, the expression can be written as $(m - ______)^2$.
Study Notes
Factorisation of Algebraic Expressions
- An expression of the form ax² + bx + c is called a quadratic trinomial.
- To find the factors of a quadratic trinomial follow these steps:
- Find the product of the coefficient of the square term and the constant term.
- Find the factors of this product that sum to the coefficient of the middle term.
- Write the middle term as the sum of these two factors.
- Group the four terms and find the common factor.
- Example: 2x² - 9x + 9 = (x - 3)(2x -3)
- Recall that:
- a² - b² = (a+b)(a-b)
- a³ + b³ = (a+b)(a² - ab + b²)
- a³ - b³ = (a-b)(a² + ab + b²)
Simplifying Rational Algebraic Expressions
- A rational algebraic expression is written in the form A/B, where A and B are algebraic expressions.
- Perform operations of addition, subtraction, multiplication, and division on rational expressions.
- Example: (2x² + 5x - 18)/(x² - 10x + 21) = (2x + 9 )(x - 2)/(x - 7)(x - 3)
Factorising a³ + b³
- (a+b)³ = a³ + 3a²b + 3ab² + b³
- a³ + b³ + 3ab(a +b) = (a + b)³
- a³ + b³ = (a + b)(a² - ab + b²)
- Example: x³ + 27y³= (x+3y)(x² -3xy + 9y²)
Practice Problems
- Factorise the following expressions:
- y³ - 27
- x³ - 64y³
- 27m³ - 216n³
- 125y³ - 1
- 8p³ - p³/27
- 343a³- 512b³
- 64x³ - 729y³
- 16a³ - b³/128
- Simplify the following expressions:
- (x + y)³ - (x - y)³
- (3a + 5b)³ - (3a - 5b)³
- (a + b)³ - a³ - b³
- p³ - (p + 1)³
- (3xy - 2ab)³ - (3xy + 2ab)³
- Factorise the following expressions:
- x² + 9x + 18
- x² - 10x + 9
- y²+ 24y + 144
- 5y² + 5y - 10
- p² - 2p - 35
- p² - 7p - 44
- m² - 23m + 120 -m² - 25m + 100 -3x² + 14x + 15 -2x² + x - 45 -20x² - 26x + 8
- 44x² - x - 3
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Description
This quiz covers the factorization of algebraic expressions, including quadratic trinomials and the simplification of rational expressions. You will explore the techniques for finding factors, grouping terms, and performing operations on rational expressions. Enhance your understanding of algebraic identities and rational expressions through this engaging quiz.
