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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 $(m - ______)^2$.
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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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