Factorisation and Rational Expressions

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Questions and Answers

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 $(x - 3)(2x - 3)$.

True

What is the product of the coefficient of the square term and the constant term in the expression $2x^2 - 9x + 9$?

The expression $p^2 - 9q^2$ can be factored as _ and _.

p + 3q, p - 3q Signup and view all the answers

Match the following expressions with their factored forms:

$x^2 - 10x + 21$ = $(x - 3)(x - 7)$ $2x^2 + 5x - 18$ = $(2x - 3)(x + 6)$ $2x^2 - 9x + 9$ = $(x - 3)(2x - 3)$ $x^2 + 4x + 4$ = $(x + 2)(x + 2)$ Signup and view all the answers

Which expression is recognized as a quadratic trinomial?

$2x^2 + 5x - 9$ Signup and view all the answers

The factors of $x^2 + (a + b)x + ab$ are $(x + a)$ and $(x + b)$.

True Signup and view all the answers

What is the sum of the two numbers that factor the quadratic $x^2 + 5x + 6$?

5 Signup and view all the answers

What is the first step in simplifying $(2x + 3y)^3 - (2x - 3y)^3$ using the difference of cubes formula?

Identify a and b in the expression. Signup and view all the answers

The difference of cubes formula can be applied to simplify the expression $(x + y)^3 - (x - y)^3$.

True Signup and view all the answers

What is the simplified form of $(x + y)^3 - (x - y)^3$?

6xy^2 Signup and view all the answers

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)][__________].

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

What is the factored form of $2y^2 - 4y - 30$?

$2(y - 5)(y + 3)$ Signup and view all the answers

Match the following algebraic expressions with their corresponding simplifications:

y^3 - 27 = (y - 3)(y^2 + 3y + 9) x^3 - 64y^3 = (x - 4y)(x^2 + 4xy + 16y^2) 125y^3 - 1 = (5y - 1)(25y^2 + 5y + 1) p^3 - (p + 1)^3 = -3p^2 - 3p - 1 Signup and view all the answers

Which of the following gives a rational algebraic expression?

\frac{2x - 3}{x + 4} Signup and view all the answers

The sum of cubes formula can be expressed as $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

True Signup and view all the answers

Dividing algebraic expressions can yield non-zero denominators.

True Signup and view all the answers

What is the first step in factoring $x^2 + 9x + 18$?

Find two numbers that multiply to 18 and add to 9. Signup and view all the answers

The expression $x^3 + 27y^3$ can be factored as $(x + 3y) [x^2 - 3xy + 9y^2]$. This is based on the __________ formula.

sum of cubes Signup and view all the answers

Factor the expression $27m^3 - 216n^3$.

27(m - 6n)(m^2 + 6mn + 36n^2) Signup and view all the answers

Match the following expressions with their factored forms:

x^2 + 9x + 18 = (x + 3)(x + 6) p^2 - 2p - 35 = (p - 7)(p + 5) 3x^2 + 14x + 15 = (3x + 5)(x + 3) 2x^2 + x - 45 = (2x - 9)(x + 5) Signup and view all the answers

Which of the following expressions can be factored as $(a + b)(a^2 - ab + b^2)$?

$a^3 + b^3$ Signup and view all the answers

The expression $x^2 - 10x + 9$ can be factored as $(x - 1)(x - 9)$.

False Signup and view all the answers

When factoring $m^2 - 25m + 100$, the expression can be written as $(m - ______)^2$.

12.5 Signup and view all the answers

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.

Factorisation and Rational Expressions

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Podcast

Questions and Answers

What are the factors of the quadratic trinomial $x^2 + 5x + 6$?

The expression $2x^2 - 9x + 9$ can be factored as $(x - 3)(2x - 3)$.

What is the product of the coefficient of the square term and the constant term in the expression $2x^2 - 9x + 9$?

The expression $p^2 - 9q^2$ can be factored as _____ and _____.

Match the following expressions with their factored forms:

Which expression is recognized as a quadratic trinomial?

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 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 $(x + y)^3 - (x - y)^3$.

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)][__________].

What is the factored form of $2y^2 - 4y - 30$?

Match the following algebraic expressions with their corresponding simplifications:

Which of the following gives a rational algebraic expression?

The sum of cubes formula can be expressed as $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Dividing algebraic expressions can yield non-zero denominators.

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.

Factor the expression $27m^3 - 216n^3$.

Match the following expressions with their factored forms:

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 $(x - 1)(x - 9)$.

When factoring $m^2 - 25m + 100$, the expression can be written as $(m - ______)^2$.

Study Notes

Factorisation of Algebraic Expressions

Simplifying Rational Algebraic Expressions

Factorising a³ + b³

Practice Problems

Studying That Suits You

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