Problem 87 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$6 w^{2}-11 w-35$$

Step-by-Step Solution

Answer

The polynomial \(6w^2 - 11w - 35\) factors to \((3w + 5)(2w - 7)\).

Step 1: Identify the Polynomial

We need to factor the quadratic polynomial \(6w^2 - 11w - 35\) completely. This is a quadratic equation of the form \(ax^2 + bx + c\), where \(a = 6\), \(b = -11\), and \(c = -35\).

Step 2: Look for a Pair of Numbers

We want two numbers that, when multiplied, give \(a \times c = 6 \times (-35) = -210\), and when added, give \(b = -11\). Let's find these numbers.

Step 3: Identify Suitable Pair

Upon checking various pairs, we find that \(10\) and \(-21\) satisfy the conditions, as \(10 \times (-21) = -210\) and \(10 + (-21) = -11\).

Step 4: Decompose the Middle Term

Rewrite the middle term \(-11w\) using the identified pair: \(6w^2 + 10w - 21w - 35\).

Step 5: Factor by Grouping

Group the terms: \((6w^2 + 10w) + (-21w - 35)\).Factor out the greatest common factor from each group.From the first group \(6w^2 + 10w\), factor out \(2w\): \(2w(3w + 5)\).From the second group \(-21w - 35\), factor out \(-7\): \(-7(3w + 5)\).

Step 6: Factor the Common Binomial

Notice the common binomial \((3w + 5)\) in the factored expressions:\(2w(3w + 5) - 7(3w + 5)\).Factor out the \((3w + 5)\):\((3w + 5)(2w - 7)\).

Step 7: Conclusion

The polynomial \(6w^2 - 11w - 35\) is factorable using integers, and its fully factored form is \((3w + 5)(2w - 7)\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra, and they appear in the form of \(ax^2 + bx + c = 0\). The term "quadratic" refers to the fact that the highest exponent is 2. Understanding how to work with quadratic equations is crucial, especially when it comes to solving them through various methods like factoring, completing the square, or using the quadratic formula.

In our specific exercise, the equation given is expressed in polynomial form but can be seen to fit a quadratic mold: \(6w^2 - 11w - 35\) . Here, the coefficient \(a\) is 6, \(b\) is -11, and \(c\) is -35.

• \(a\) affects the curve's direction and width in the graph of the equation.
• \(b\) is responsible for the linear component, determining the slope at any point.
• \(c\) is the constant term, shifting the entire graph up or down.
  Understanding how these coefficients interact is key to solving and factoring quadratic equations.

Polynomial Factorization

Factoring polynomials involves breaking down a complex expression into simpler ones. This is an essential skill that simplifies equations, making them easier to work with or solve. Think of polynomial factorization as unwrapping a tricky package or solving a puzzle to reveal its components.

The polynomial given, \(6w^2 - 11w - 35\), is factored by expressing it as a product of simpler binomials. The goal is to rewrite it in a form like \((ax + b)(cx + d)\), where the product gives the original polynomial.

• Each pair \((ax + b)\) and \((cx + d)\) are carefully chosen such that their product equals the original polynomial.
• This involves identifying the right numbers that satisfy both the multiplication and addition conditions of the coefficients.
• For this example, the factors \((3w + 5)(2w - 7)\) satisfy the requirements, resulting from their product \(6w^2 - 11w - 35\).
  Mastering polynomial factorization often simplifies solving even the most complex algebraic puzzles.

Factoring by Grouping

Factoring by grouping is a strategic method of polynomial factorization used when other techniques may not be immediately obvious. It is incredibly useful in handling polynomials that do not readily present "nice" factors or forms.

In our example, \(6w^2 - 11w - 35\), the middle term needs to be split to facilitate grouping. This technique involves:

• **Identifying pairs of terms**: Create sub-expressions that share a common factor.
• **Factoring each group**: Determine the greatest common factor (GCF), then factor it out from each set of terms, creating a unified common binomial factor across terms.
• For the polynomial at hand, we group and factor as: - Group 1: \(6w^2 + 10w\) becomes \(2w(3w + 5)\) - Group 2: \(-21w - 35\) becomes \(-7(3w + 5)\)
• Successfully finish by factoring out the common binomial \((3w + 5)\) to achieve the factored form, \((2w - 7)(3w + 5)\).
  This makes the grouping method an elegant tool when factoring more complex polynomials not immediately yielding to simpler methods.