Answered

Which of the following statements about $42 x y-49 x+30 y-35$ are true? Check all that apply.


  • One of the factors is $(6 y+7)$.
  • One of the factors is $(7 x+5)$.
  • One of the factors is $(6 y-7)$.
  • One of the factors is $(7 x-5)$.
  • This expression is prime.

Verified Answer :

  • Group the terms of the polynomial: \((42xy - 49x) + (30y - 35)\).
  • Factor out the greatest common factor from each group: \(7x(6y - 7) + 5(6y - 7)\).
  • Extract the common binomial factor to express the polynomial as a product: \((7x + 5)(6y - 7)\).
  • Conclude that the true statements are: \(\boxed{\text{One of the factors is } (7x + 5) \text{ and one of the factors is } (6y - 7)}\).

Explanation

  1. Problem Analysis

The objective is to factor the expression \(42xy - 49x + 30y - 35\). A useful technique for a polynomial with four terms is "factoring by grouping," where terms are paired to identify common binomial factors. The expression has four terms: \(42xy\), \(-49x\), \(30y\), and \(-35\). By grouping the terms properly, we can simplify the expression into a product of two linear binomials.

  1. Factoring by Grouping

To factor the expression \(42xy - 49x + 30y - 35\), we group the first two terms together and the last two terms together:
\[(42xy - 49x) + (30y - 35)\]
Now, we factor out the greatest common factor (GCF) from each group:

  • From the first group \((42xy - 49x)\), the GCF is \(7x\), giving \(7x(6y - 7)\).
  • From the second group \((30y - 35)\), the GCF is \(5\), giving \(5(6y - 7)\).

Substituting these back into our expression, we get:
\[7x(6y - 7) + 5(6y - 7)\]
Since \((6y - 7)\) is common to both terms, we can factor it out:
\[(7x + 5)(6y - 7)\]
The expression factors into \((7x + 5)\) and \((6y - 7)\).

  1. Evaluating the Statements

We have factored the expression \(42xy - 49x + 30y - 35\) into \((7x + 5)(6y - 7)\). Now, we evaluate the given options:

  • "One of the factors is \((6y + 7)\)": This is false, as the factor is \((6y - 7)\).
  • "One of the factors is \((7x + 5)\)": This is true.
  • "One of the factors is \((6y - 7)\)": This is true.
  • "One of the factors is \((7x - 5)\)": This is false.
  • "This expression is prime": This is false, as it can be factored.

Therefore, the true statements are that one of the factors is \((7x + 5)\) and one of the factors is \((6y - 7)\).

Examples

Factoring polynomials is essential for simplifying complex equations and solving for variables in fields like engineering and physics. For example, when calculating the area of a rectangle whose sides are defined by algebraic expressions like \((7x+5)\) and \((6y-7)\), you need to be able to distribute or factor them to determine the total area or find the dimensions if the area is known. Mastery of factoring by grouping allows you to work backwards from a complex area expression to the fundamental dimensions of the shape.