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Chapter 6: Problem 11

Factor: \(3 x^{2}(4 x-7)+8(4 x-7)\).

Short Answer

Expert verified
The expression factors to \((4x-7)(3x^2 + 8)\).

Step by step solution

01

Identify the Common Factor

Start by looking at the expression: \(3x^{2}(4x-7) + 8(4x-7)\). Notice that \((4x-7)\) appears in both terms. This means \((4x-7)\) is a common factor.
02

Factor Out the Common Factor

Factor \((4x-7)\) out of the expression. Writing the expression inside the parentheses, we get: \((4x-7)(3x^2 + 8)\).
03

Verify the Factored Expression

Multiply \((4x-7)\) by \((3x^2 + 8)\) to make sure it returns the original expression: \((4x-7)(3x^2) + (4x-7)(8)\) = \(3x^{2}(4x-7) + 8(4x-7)\). This confirms our factored form is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor
In algebra, spotting the common factor in an expression is like finding a hidden key that simplifies the entire equation. When you are given a complex expression, look for terms that appear in multiple parts of the expression. These repeating parts are your clues.

In our example, the expression is \(3x^2(4x-7) + 8(4x-7)\). Here, \((4x-7)\) is a term present in both parts of the expression. This repetition indicates that \((4x-7)\) is the common factor.

  • Always inspect the entire expression carefully to identify common factors.
  • Highlight or underline common parts to help visualize them better.
  • Using the common factor can often dramatically simplify an expression or help in further algebraic manipulations.
Factoring Techniques
Factoring techniques are strategies used to break down polynomials into simpler components. Understanding and applying these techniques correctly can make solving polynomial expressions much easier. The principal idea is to express the polynomial as a product of its factors.

In our exercise, after identifying \((4x-7)\) as the common factor, we factor it out. This involves writing the common term outside a pair of parentheses, leaving inside the parentheses the multiplied terms that made up the original expression. So, \(3x^2(4x-7) + 8(4x-7)\) becomes \((4x-7)(3x^2 + 8)\).
  • Once the common factor is spotted, use it to construct a simpler expression.
  • The aim is to "factor out" the common parts to reveal underlying simpler terms.
  • These techniques not only simplify the expression but also prepare it for further operations, such as solving or graphing.
Polynomial Expressions
Polynomials are algebraic expressions that consist of variables and coefficients, structured in terms involving powers of the variables. They are fundamental in math due to their straightforward and predictable properties.

The expression in our example, \(3x^2(4x-7) + 8(4x-7)\), is a polynomial. It includes two main terms, each multiplied by the polynomial \((4x-7)\). Such expressions are ubiquitous in algebra, serving as building blocks for more complex equations.
  • Each term in a polynomial is composed of constants (coefficients) and variables raised to non-negative integer powers.
  • The degree of a polynomial is the highest power of the variable present in the expression.
  • Polynomials can be factored, expanded, and manipulated according to algebraic rules, making them versatile tools in problem-solving.

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Most popular questions from this chapter

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