Practice Questions

Name the identity used to factorise an expression of the form $p^2 - q^2$.

Define the term 'irreducible factor' in the context of algebraic expressions.

Solve by division: $48p^4 \div (-8p^2)$.

Explain what is meant by the 'prime factor form' of a natural number, using the number 90 as an example.

Justify why the expression $5x(y+3)$ is considered to be in its irreducible factor form, while $5(xy+3x)$ is not.

What is the greatest common factor of the terms $9x^2y$ and $21xy^2$?

Examine the expression $x^2 + 7x + 10$ and find two numbers whose product is 10 and sum is 7.

Factorise the expression: $15x - 60$.

A student claims that since $x^2 - 4 = (x-2)(x+2)$, then $x^2+4$ must equal $(x+2)(x+2)$. Critique this claim.

Find the greatest common factor of the terms $14a^2b$ and $21ab^2c$.

Evaluate whether rearranging the terms of $z - 7 + 7xy - xyz$ to $7xy - 7 - xyz + z$ is a valid first step for factorisation by regrouping. Justify your answer.

Factorise the expression $7x^2y - 21xy^2 + 28xy$.

Design a polynomial with four terms which, when correctly regrouped, results in the factors $(2x^2 + 5)$ and $(3y - 4)$. Demonstrate the factorisation process for your designed polynomial.

The area of a square is given by the expression $A = 49y^2 - 84yz + 36z^2$ square units. a) Formulate an expression for the side length of the square. b) Justify that the given expression must represent a perfect square. c) Design an expression for the perimeter of the square. d) If $y=5$ and $z=3$, evaluate the area and perimeter.

Create a division problem where the dividend is $p^4 - 81$ and the quotient is $(p^2+9)$. Determine the divisor and justify your answer through complete factorisation.

A student simplifies $\frac{x^3 + 2x^2 + 3x}{2x}$ to $x^2 + x + 3$. Critique this simplification, identify the error, and provide the correct answer.

Create a trinomial of the form $x^2+px+q$ that has $(x-8)$ as a factor and where the constant term $q$ is $24$. Formulate the complete expression and show its factorisation.

Factorise by regrouping terms: $12ab - 8a + 9b - 6$.

List all the factors of the algebraic term $14ab^2$.

Recall the expanded form of the product $(x+a)(x+b)$.

Identify the common factors in the terms of the expression $8x^2y^2z, 12x^3y, 16xy^3$.

Describe the first step in factorising the expression $5ab + 10a$ using the method of common factors.

Apply a suitable identity to factorise $25m^2 + 90mn + 81n^2$.

Solve the division: $(16x^3 - 4x^2 + 8x) \div (4x)$.

Explain why $3 \times (mn)$ is not considered the irreducible form of $3mn$.

Explain the 'method of common factors' for factorising an algebraic expression. Use the expression $10a^2 - 15b^2 + 20c^2$ to illustrate your explanation.

Factorise the expression $p^2 - 12p - 45$.

Factorise completely: $a^4 - 16b^4$.

Describe the relationship between the multiplication and division of algebraic expressions.

Calculate the value of $(102)^2 - (98)^2$ using a suitable identity.

Critique the following statement: 'To factorise $x^2 + 9$, we can write it as $(x)^2 + (3)^2$, so the factors are $(x+3)(x+3)$.' Pinpoint the error in reasoning.

Propose a binomial that must be added to the expression $p^4 - 18p^2 + 81$ to transform it into $(p^2+9)^2$.

Factorise and then divide: $(x^2 - 9x + 20) \div (x-4)$.

Evaluate the expression $(x^4 - (x-z)^4)$ by factorising it completely. Justify your use of the difference of squares identity at each step.

Justify, using factorisation, why the division of $25a^2 - 4b^2 + 28bc - 49c^2$ by $(5a - 2b + 7c)$ results in $(5a + 2b - 7c)$.

List the factors of the expression $7(p+q)(r+s)$. Are these factors irreducible? Explain why or why not.

Formulate a proof to demonstrate that the product of any four consecutive integers is always one less than a perfect square. Use factorisation and regrouping to justify your conclusion.

The area of a rectangular field is given by the expression $(6x^2 + 13x - 5)$ square units. If the width is $(2x+5)$ units, analyze the expression to find the length of the field.

Summarize how to check if a trinomial of the form $x^2 + px + q$ can be factorised using the identity $(a+b)^2 = a^2+2ab+b^2$.

Factorise and then perform the division: $44p^3(18p^2 - 50) \div 22p^2(3p - 5)$.

Consider the expression $P(x) = x(x+1)(x+2)(x+3) \div x(x+1)$. a) A student simplifies this by cancelling $x$ and $(x+1)$ to get $(x+2)(x+3)$. Justify for which values of $x$ this simplification is not valid. b) Create a new expression, $Q(x)$, which is equivalent to $P(x)$ for all values of $x$ except the invalid ones, but is defined for those values. c) Evaluate the difference between the sum of factors of $x^2+5x+6$ and the sum of factors of $x^2-5x+6$.

Formulate a general rule for factorising an expression of the form $a^4 + 4b^4$. (Hint: Try adding and subtracting a term to create a perfect square). Validate your rule by factorising $x^4 + 64$.

Describe the process of 'factorisation by regrouping terms'. Explain the steps using the expression $15xy - 6x + 5y - 2$.

Factorise the expression $50a^2 - 98b^2$.

Summarize the steps to divide a polynomial by a polynomial, for example, $(y^2 + 7y + 10) \div (y+5)$. Explain why we cannot just divide each term of the dividend by the divisor.