Answer :
- Factor out the common factor $x$ from $x^2 - 3x$, resulting in $x(x-3)$.
- Factor out the common factor $6x$ from $6x^3 + 18x$, resulting in $6x(x^2 + 3)$.
- Factor out the common factor $3x$ from $3x^3 - 12x$, resulting in $3x(x^2 - 4)$, then factor the difference of squares to get $3x(x-2)(x+2)$.
- Factor out the common factor $x^2$ from $7x^2 + 25x^3$, resulting in $x^2(7 + 25x)$.
- The fully factorised expressions are $x(x - 3)$, $6x(x^2 + 3)$, $3x(x - 2)(x + 2)$, and $x^2(7 + 25x)$.
$\boxed{x(x - 3), 6x(x^2 + 3), 3x(x - 2)(x + 2), x^2(7 + 25x)}$
### Explanation
1. Introduction
We are given four expressions to factorise fully. Let's factorise each one step by step.
2. Factorising the first expression
1. **Factorising $x^2 - 3x$:**
The common factor in both terms is $x$. Factoring out $x$, we get:
$$x^2 - 3x = x(x - 3)$$
This is the fully factorised form.
3. Factorising the second expression
2. **Factorising $6x^3 + 18x$:**
The common factor in both terms is $6x$. Factoring out $6x$, we get:
$$6x^3 + 18x = 6x(x^2 + 3)$$
This is the fully factorised form.
4. Factorising the third expression
3. **Factorising $3x^3 - 12x$:**
The common factor in both terms is $3x$. Factoring out $3x$, we get:
$$3x^3 - 12x = 3x(x^2 - 4)$$
Now, we can see that $x^2 - 4$ is a difference of squares, which can be factorised as $(x - 2)(x + 2)$. Therefore,
$$3x(x^2 - 4) = 3x(x - 2)(x + 2)$$
This is the fully factorised form.
5. Factorising the fourth expression
4. **Factorising $7x^2 + 25x^3$:**
The common factor in both terms is $x^2$. Factoring out $x^2$, we get:
$$7x^2 + 25x^3 = x^2(7 + 25x)$$
This is the fully factorised form.
6. Final Answer
In summary, the fully factorised forms of the given expressions are:
1. $x^2 - 3x = x(x - 3)$
2. $6x^3 + 18x = 6x(x^2 + 3)$
3. $3x^3 - 12x = 3x(x - 2)(x + 2)$
4. $7x^2 + 25x^3 = x^2(7 + 25x)$
### Examples
Factoring is a fundamental skill in algebra and is used extensively in various real-world applications. For instance, engineers use factoring to simplify complex equations when designing structures or analyzing systems. Similarly, economists use factoring to model and predict economic trends. In computer science, factoring is used in cryptography and data compression algorithms. Understanding factoring helps in simplifying problems and finding efficient solutions across many disciplines.
- Factor out the common factor $6x$ from $6x^3 + 18x$, resulting in $6x(x^2 + 3)$.
- Factor out the common factor $3x$ from $3x^3 - 12x$, resulting in $3x(x^2 - 4)$, then factor the difference of squares to get $3x(x-2)(x+2)$.
- Factor out the common factor $x^2$ from $7x^2 + 25x^3$, resulting in $x^2(7 + 25x)$.
- The fully factorised expressions are $x(x - 3)$, $6x(x^2 + 3)$, $3x(x - 2)(x + 2)$, and $x^2(7 + 25x)$.
$\boxed{x(x - 3), 6x(x^2 + 3), 3x(x - 2)(x + 2), x^2(7 + 25x)}$
### Explanation
1. Introduction
We are given four expressions to factorise fully. Let's factorise each one step by step.
2. Factorising the first expression
1. **Factorising $x^2 - 3x$:**
The common factor in both terms is $x$. Factoring out $x$, we get:
$$x^2 - 3x = x(x - 3)$$
This is the fully factorised form.
3. Factorising the second expression
2. **Factorising $6x^3 + 18x$:**
The common factor in both terms is $6x$. Factoring out $6x$, we get:
$$6x^3 + 18x = 6x(x^2 + 3)$$
This is the fully factorised form.
4. Factorising the third expression
3. **Factorising $3x^3 - 12x$:**
The common factor in both terms is $3x$. Factoring out $3x$, we get:
$$3x^3 - 12x = 3x(x^2 - 4)$$
Now, we can see that $x^2 - 4$ is a difference of squares, which can be factorised as $(x - 2)(x + 2)$. Therefore,
$$3x(x^2 - 4) = 3x(x - 2)(x + 2)$$
This is the fully factorised form.
5. Factorising the fourth expression
4. **Factorising $7x^2 + 25x^3$:**
The common factor in both terms is $x^2$. Factoring out $x^2$, we get:
$$7x^2 + 25x^3 = x^2(7 + 25x)$$
This is the fully factorised form.
6. Final Answer
In summary, the fully factorised forms of the given expressions are:
1. $x^2 - 3x = x(x - 3)$
2. $6x^3 + 18x = 6x(x^2 + 3)$
3. $3x^3 - 12x = 3x(x - 2)(x + 2)$
4. $7x^2 + 25x^3 = x^2(7 + 25x)$
### Examples
Factoring is a fundamental skill in algebra and is used extensively in various real-world applications. For instance, engineers use factoring to simplify complex equations when designing structures or analyzing systems. Similarly, economists use factoring to model and predict economic trends. In computer science, factoring is used in cryptography and data compression algorithms. Understanding factoring helps in simplifying problems and finding efficient solutions across many disciplines.
