Question

Expressions that occur in calculus are given. Factor each expression completely. $$ 3 x^{2}(8 x-3)+x^{3} \cdot 8 $$

Short answer

The factored expression is \[x^2(32x - 9)\].

Step-by-step solution

Distribute and Simplify

The given expression is \[3x^2(8x-3) + x^3 \times 8\]. Start by distributing the terms: \[3x^2 \times 8x - 3x^2 \times 3 + x^3 \times 8\]. This gives us: \[24x^3 - 9x^2 + 8x^3\].

Combine Like Terms

Combine the like terms: \[24x^3 + 8x^3 - 9x^2\] which simplifies to: \[32x^3 - 9x^2\].

Factor Out the Greatest Common Factor

Identify the greatest common factor (GCF). The GCF of \[32x^3\] and \[-9x^2\]is \[x^2\]. Factor out \[x^2\] from the expression: \[x^2 (32x - 9)\].

Key concepts

distributive property

The distributive property helps us simplify algebraic expressions by distributing a single term across terms inside parentheses.
In our problem, we start with the expression \(3x^2(8x-3) + x^3 \cdot 8\).
To distribute, we multiply each term inside the parentheses by the term outside: \(3x^2 \times 8x - 3x^2 \times 3 + x^3 \times 8\).
This action helps break down the expression into simpler parts we can work with.
The result of distribution here is \(24x^3 - 9x^2 + 8x^3\).
Using the distributive property is a fundamental step in simplifying complex algebraic expressions.

combining like terms

Combining like terms is the process of adding or subtracting terms that have the same variable to a certain power.
In our simplified expression from the previous step, \(24x^3 - 9x^2 + 8x^3\), we notice terms that can be combined.
The terms \(24x^3\) and \(8x^3\) are like terms because they both contain \(x^3\).
We add these coefficients together: \(24 + 8 = 32\).
This combines to \(32x^3\).
We are then left with the expression: \(32x^3 - 9x^2\).
Combining like terms helps consolidate our expression into a simpler form, making it easier to factor further.

greatest common factor

The Greatest Common Factor (GCF) is the highest factor that divides two or more terms.
For the expression \(32x^3 - 9x^2\), we need to identify the GCF of \(32x^3\) and \(-9x^2\).
The GCF must include the highest common power of \(x\) present in both terms, which is \(x^2\), since both terms have \(x^2\) as a factor.
We then factor out \x^2 from the expression: \(x^2(32x - 9)\).
This leaves us with a simpler factored expression.
Understanding how to find and factor out the GCF is crucial because it simplifies and reduces the complexity of algebraic expressions, making solving these expressions easier.