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Chapter 6: Problem 55

Factor completely. If a polynomial is prime, state this. $$ 8 m^{3} n-32 m^{2} n^{2}+24 m n $$

Short Answer

Expert verified
Factor completely: \(8mn (m - 3n)(m - n)\)

Step by step solution

01

Identify the common factor

Examine each term in the polynomial and find the greatest common factor (GCF). For the polynomial \(8m^3n - 32m^2n^2 + 24mn\), the GCF is \(8mn\). Terms: \(8m^3n \rightarrow 8mn \times m^2$$-32m^2n^2 \rightarrow 8mn \times (-4mn)\) \(+24mn \rightarrow 8mn \times 3\)
02

Factor out the GCF

Factor the GCF \(8mn\) from each term in the polynomial:\(8mn (m^2 - 4mn + 3)\)
03

Factor the remaining polynomial

Next, factor the quadratic polynomial \(m^2 - 4mn + 3\). Look for two numbers that multiply to \(3\) and add to \(-4mn\). These numbers are \(-3mn\) and \(-mn\). Therefore, the quadratic can be factored as:\((m - 3n)(m - n)\)
04

Combine the factors

Combine the GCF and the factored quadratic polynomial:\(8mn (m - 3n)(m - n)\)

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor (GCF) is the largest factor that divides every term in a polynomial without leaving a remainder. It simplifies polynomials by reducing their overall degree, making them easier to factor.
In the exercise, we identified the GCF of the polynomial \(8m^3n - 32m^2n^2 + 24mn\). Here鈥檚 how we found it:
  • First, look at each term: \(8m^3n\), \(-32m^2n^2\), and \(24mn\).
  • Identify the common numerical factor; it鈥檚 \(8\) for all terms.
  • Next, check the variable part. The smallest power of \(m\) is \(m\), and for \(n\) it's \(n\) itself.
So, the GCF for this polynomial is \(8mn\). Factoring out the GCF simplifies the polynomial to \(8mn(m^2 - 4mn + 3)\).
Quadratic Polynomial
A quadratic polynomial is a second-degree polynomial in the form \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants. Quadratic polynomials often need to be factored to solve related algebra problems.
In our exercise, once we factored out the GCF \(8mn\), we had the quadratic polynomial \(m^2 - 4mn + 3\). To factor it, we need to find two numbers that:
  • Multiply to the constant term, \(3\).
  • Add to the coefficient of the middle term, \(-4n\).
The numbers that work are \(-3n\) and \(-n\), since \((m - 3n)(m - n)\) multiply to \(3mn\) and add up to \(-4mn\). This is the factorization method usually required for solving quadratic polynomials.
Factoring Techniques
Factoring is breaking down a complex expression into simpler components or factors that, when multiplied together, produce the original expression. Different techniques can be applied based on the structure of the polynomial:
  • Factoring out the GCF: As explained, this simplifies the polynomial by reducing its degree.
  • Factoring quadratics: Look for two numbers that multiply to the constant term and add to the coefficient of the middle term.
  • Group factoring: For polynomials with four or more terms, grouping terms can reveal common factors.
In our specific example, the technique involved: identifying the GCF, then factoring a quadratic polynomial. The final factored form was \(8mn(m - 3n)(m - n)\).

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Most popular questions from this chapter

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