Factorise this expression as fully as possible
4x² + x3
The Correct Answer and Explanation is:
To factorise the expression (4x^2 + x^3), follow these steps:
Step 1: Identify the common factor
First, examine the terms in the expression:
- The first term is (4x^2).
- The second term is (x^3).
Both terms contain a factor of (x^2). We can factor out (x^2) from both terms.
Step 2: Factor out (x^2)
When you factor out (x^2) from each term, you get:
[
4x^2 + x^3 = x^2(4 + x)
]
Step 3: Verify the factorisation
To check if this factorisation is correct, you can distribute (x^2) back into the terms in the parentheses:
[
x^2(4 + x) = 4x^2 + x^3
]
Since the original expression matches this result, the factorisation is correct.
Final Answer:
The fully factorised form of (4x^2 + x^3) is:
[
x^2(4 + x)
]
Explanation:
In factorisation, the goal is to express the given algebraic expression as a product of simpler expressions. In this case, the expression (4x^2 + x^3) contains two terms, and both terms share a common factor of (x^2). By factoring out this common factor, we simplify the expression to a product of (x^2) and the remaining binomial (4 + x).
Factorisation is particularly useful in solving equations, simplifying expressions, and analyzing polynomial relationships. The process typically involves identifying common factors, applying distributive properties, and reducing the expression to its simplest form. In this case, factoring out the greatest common factor (GCF) (x^2) allowed us to break down the expression into a more manageable form.
This type of factorisation is straightforward but is an essential skill when working with polynomials in algebra, as it lays the foundation for more complex operations, such as solving quadratic equations, simplifying rational expressions, or finding roots of polynomials.