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Chapter 6: Problem 114

Is \(5 x(x+2)-3(x+2)\) fully factored? Explain.

Short Answer

Expert verified
Yes, \(5x(x+2)-3(x+2)\) is fully factored as \((x+2)(5x-3)\).

Step by step solution

01

Identify Common Factors

Examine the expression \(5x(x+2) - 3(x+2)\) to identify any common factors. Both terms \(5x(x+2)\) and \(-3(x+2)\) share the factor \((x+2)\).
02

Factor Out the Greatest Common Factor

Since \((x+2)\) is common to both terms, factor it out of the expression. This gives us: \((x+2)(5x - 3)\).
03

Verify Factored Form

Check if the expression \((x+2)(5x - 3)\) can be further factored by examining each part. Both \((x+2)\) and \((5x - 3)\) are irreducible over the integers, meaning the expression is fully factored.

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Key Concepts

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

Understanding Common Factors
In algebra, a common factor is a number or an algebraic expression that divides each term in a given expression without leaving a remainder. Identifying common factors in an expression is a crucial step when simplifying and factoring polynomials. In the provided exercise, the expression \(5x(x+2) - 3(x+2)\) has been analyzed for commonality.

  • Both terms, \(5x(x+2)\) and \(-3(x+2)\), included the factor \((x+2)\). The presence of a common factor makes it possible to combine or simplify the expression.
  • Factoring out this common factor \((x+2)\) simplifies the expression, revealing the algebraic structure more clearly.
  • By understanding common factors, you can also make calculations easier by reducing expressions to their simplest form before performing arithmetic operations.
Greatest Common Factor (GCF)
The "Greatest Common Factor" (GCF) is the largest factor that two or more expressions share. It鈥檚 an essential concept used to reduce expressions in algebra.

In the step-by-step solution for the expression \(5x(x+2) - 3(x+2)\), the GCF was determined to be \((x+2)\). This is because:

  • Among the terms \(5x(x+2)\) and \(-3(x+2)\), \((x+2)\) is the only factor that appears in both.
  • Factoring out the GCF, the expression simplifies as \((x+2)(5x - 3)\). The GCF provides not just a simpler form but a way to check that no other common terms were overlooked.
Understanding the GCF helps ensure you're fully factoring expressions, making subsequent steps like solving equations more straightforward.
Irreducible Polynomials
An irreducible polynomial is one that cannot be factored down further over a given number set or field, such as integers, rationals, or reals. In simple terms, these are polynomials that are already in their simplest form.

In the expression \((x+2)(5x - 3)\):

  • The polynomials \((x+2)\) and \((5x - 3)\) are irreducible because they cannot be factored further within the integers.
  • This means no matter how we analyze them, there are no other number pairs whose multiplication gives these expressions over integers.
  • Recognizing irreducible polynomials in a factored expression ensures that the factorization process is complete, which is crucial when solving polynomial equations.

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