Question 1

What is the common factor of $3x$ and $15$?

Accepted Answer

To find the common factor of $3x$ and $15$, we find the factors of each term. The factors of $3x$ are $1, 3, x,$ and $3x$. The factors of $15$ are $1, 3, 5,$ and $15$. The common factor present in both is $3$. Therefore, the correct option is $3$.

Question 2

Factorise the expression: $7x - 42$

Accepted Answer

To factorise $7x - 42$, we look for the highest common factor of the terms $7x$ and $42$. The number $42$ can be written as $7 	imes 6$. Thus, the expression becomes $7x - 7 	imes 6$. Taking $7$ common, we get $7(x - 6)$. Therefore, the correct option is $7(x - 6)$.

Question 3

What are the factors of $a^2 - b^2$?

Accepted Answer

The expression $a^2 - b^2$ is a standard algebraic identity representing the difference of two squares. According to the identity, $a^2 - b^2 = (a - b)(a + b)$. Therefore, the correct option is $(a + b)(a - b)$.

Question 4

Factorise: $p^2 + 6p + 9$

Accepted Answer

The given expression is $p^2 + 6p + 9$. We can write it as $p^2 + 2(p)(3) + 3^2$. This is of the form $a^2 + 2ab + b^2 = (a + b)^2$. Here, $a = p$ and $b = 3$. Thus, it factorises to $(p + 3)(p + 3)$. Therefore, the correct option is $(p + 3)^2$.

Question 5

Find the common factors of the terms: $2xy, 3x^2, 2$

Accepted Answer

The terms are $2xy$, $3x^2$, and $2$. Their numerical and literal factors are: Factors of $2xy$ are $1, 2, x, y, 2x, 2y, xy, 2xy$; factors of $3x^2$ are $1, 3, x, x^2, 3x, 3x^2$; factors of $2$ are $1, 2$. The only common factor in all three terms is $1$. Therefore, the correct option is $1$.

Factorisation