Question

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

Short answer

(4x - 3)(12x - 3)

Step-by-step solution

Identify Common Factors

Look for common factors in the terms of the expression. In this case, both terms in the expression contain \(4x - 3\). Recognize that you can factor out this common term.

Rewrite the Expression

Rewrite the given expression with the common factor outside: \[ (4x - 3)^{2} + x \cdot 2(4x - 3) \cdot 4 \] can be rearranged to \[ (4x - 3)^{2} + 8x(4x - 3) \]

Factor Out the Common Term

Factor out the common term \(4x - 3\) from both terms: \[ (4x - 3)[(4x - 3) + 8x] \]

Simplify Inside the Brackets

Simplify the expression inside the brackets: \[ (4x - 3)[4x - 3 + 8x] = (4x - 3)[12x - 3] \]

Key concepts

common factors

When working with algebraic expressions, identifying common factors is essential. A common factor is an element that is shared by all terms in an expression. For example,
in the given exercise, the common factor is \((4x - 3)\). Recognizing common factors helps simplify expressions and makes them easier to work with.
You can find common factors by:

• Inspecting each term carefully
• Identifying any repeated elements
• Factoring out these elements

In this case, both terms \((4x - 3)^{2}\) and \(8x(4x - 3)\) contain the factor \((4x - 3)\). By factoring this out, you simplify the expression to
\((4x - 3)[(4x - 3) + 8x]\). The benefit of identifying common factors is greatly visible as it simplifies further calculations.

factoring techniques

Factoring techniques are methods used to simplify expressions or solve equations. In the given problem, factoring was used to rewrite and simplify the expression. The primary techniques include:

• Factoring out the greatest common factor (GCF)
• Using special factoring formulas like difference of squares and perfect square trinomials
• Grouping terms to find common factors

In our exercise, the technique used was to first identify and group the common factor \((4x-3)\). Once identified, we rewrote the expression as
\[(4x-3)((4x-3) + 8x)\]. This then makes it easier to simplify further, breaking down the problem into more manageable pieces. Mastering these techniques is crucial for solving algebra problems efficiently.

polynomials

Polynomials are expressions involving variables raised to non-negative integer powers and coefficients. They are the cornerstone of algebra and calculus.
For example, both terms in \[(4x-3)^2 + 8x(4x - 3)\] are polynomials. The degrees and coefficients of these polynomials guide us in identifying common factors and factoring techniques.

• Polynomials can have different degrees, such as linear, quadratic, cubic, etc.
• Working with polynomials often involves combining like terms and simplifying expressions

In this example, we dealt with quadratic and linear polynomials. By factoring, you simplify the polynomial to a form that reveals its structure
more clearly:
\((4x-3)(12x-3)\). Understanding polynomials and how to manipulate them is key to mastering higher math concepts.