Question

Expressions that occur in calculus are given. Factor each expression completely. $$ 3 x^{2}(3 x+4)^{2}+x^{3} \cdot 2(3 x+4) \cdot 3 $$

Short answer

The factored expression is \( 3x^{2}(3x+4)(5x+4) \).

Step-by-step solution

Identify the common factor

Look for common factors in both terms of the given expression. The expression is: \[ 3x^{2}(3x+4)^{2} + x^{3} \times 2(3x+4) \times 3 \] Notice that \( (3x+4) \) appears in both terms.

Factor out the greatest common factor

Extract the greatest common factor (GCF) from both terms. The GCF in this case is \( 3x^{2}(3x+4) \). Let’s factorize the expression using the GCF: \[ 3x^{2}(3x+4) \big [ (3x+4) + x \times 2 \big ] \] This simplifies the original expression.

Simplify the remaining expression inside the brackets

Now, simplify the term inside the brackets: \[ (3x+4) + 2x \] Combine like terms: \[ 3x + 4 + 2x = 5x + 4 \] Thus, the factored form is: \[ 3x^{2}(3x+4)(5x+4) \]

Key concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the highest factor that divides two or more terms. Finding the GCF is a crucial step in factoring algebraic expressions, as it simplifies the expression and makes it easier to work with.

To identify the GCF, look for common elements in each term. For example, in the expression \(3x^{2}(3x+4)^{2} + x^{3} \cdot 2(3x+4) \cdot 3\), we see that \(3x+4\) is present in both terms.
Note: Factors are numbers or variables that are multiplied together to get another number or expression.

• Identify the common variables and their lowest power.
• Identify the common numerical coefficients.

These common elements make up the GCF, which can then be factored out from the expression.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler 'factor' expressions that, when multiplied together, give the original polynomial. This process simplifies solving equations and understanding the polynomial's behavior.

Starting with the expression from the exercise: \[ 3x^{2}(3x+4)^{2} + x^{3} \times 2(3x+4) \times 3 \] We identify the GCF, \(3x^{2}(3x+4)\). Factoring this out, we rewrite the expression as: \[ 3x^{2}(3x+4) \big [ (3x+4) + x \times 2 \big ] \] This step reveals the remaining terms inside the brackets, \( (3x+4) + 2x \).

Key points to remember:

• Factoring makes complex expressions easier to handle.
• Always verify by expanding to ensure you get back the original polynomial.

Algebraic Simplification

Algebraic simplification is the process of reducing complex expressions to their simplest form. This makes them easier to understand and solve.

In our exercise, after factoring out the GCF, we had the term inside the brackets: \( (3x+4) + 2x \). Simplifying further, we combined like terms: \[ (3x + 4 + 2x = 5x + 4) \] Now the expression is simplified.

Steps for simplifying:

• Combine like terms (terms with the same variable and exponent).
• Reduce fractions if possible.
• Factor terms where possible.

Simplification helps in solving equations and understanding the relationships between terms.

Combining Like Terms

Combining like terms means adding or subtracting terms that have the same variable and exponent. This is a fundamental skill in algebra as it simplifies expressions.

From the expression \( (3x + 4) + 2x \) in our exercise, we combined the terms with the variable 'x': \[ 3x + 2x = 5x \] So, \[ (3x + 4) + 2x = 5x + 4 \] Combining like terms reduces the complexity of an expression and helps in further steps of solving equations or factoring.

Tips:

• Only combine terms with the same exact variable and exponent.
• Do not combine terms that have different variables or exponents.
• Rearrange terms if necessary to bring like terms together.

Mastering this concept is crucial in algebra as it lays the foundation for more advanced topics.