Problem 44 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$35 w^{2}-2 w-1$$

Step-by-Step Solution

Answer

The complete factorization of the given trinomial \(35w^{2}-2w-1\) is \((5w-1)(7w+1)\).

Step 1: Identify a common factor

Look at all the terms in the trinomial to see if they share a common factor. In this case, the terms do not share a common factor. Therefore, proceed to next step.

Step 2: Factor using special patterns

There are special patterns one can use to factor some trinomials, such as factoring a difference of squares or a perfect square trinomial. In this case, the trinomial does not fit any special patterns, proceed to the next step.

Step 3: Factor using the AC method

The AC method involves multiplying the coefficient of the \(w^{2}\) term, which is \(35\) by the constant term, which is \(-1\). Therefore, -35 is the product. Now, look for two numbers that multiply to -35 and add up to -2. The numbers -7 and 5 satisfy this. Therefore, the trinomial can be rewritten as: \(35w^{2} -7w + 5w -1\). Then, the trinomial can be broken into two groups, yielding: \(35w^{2} -7w \) and \(+ 5w -1\). Factor out the greatest common factor from each group. The groups are then factored to \(7w(5w -1) + 1(5w -1)\). Since the terms in parentheses are identical, they can be written as one term, resulting in \((5w -1)(7w +1)\).

Step 4: Check the solution

Multiply out \((7w+1)(5w-1)\) to confirm that this equals the original trinomial. Yes, \((5w-1) * (7w+1) = 35w^{2} - 2w -1\). The factorization of the trinomial is correct.

Key Concepts

Trinomial

A trinomial is a type of polynomial that contains exactly three terms. It is important to recognize a trinomial as it allows us to apply specific factoring techniques tailored for three-term expressions. In the given problem, the trinomial is expressed as \(35w^2 - 2w - 1\). This expression consists of:

• A quadratic term, \(35w^2\), where the variable \(w\) is raised to the power of 2.


• A linear term, \(-2w\), where \(w\) is raised to the power of 1.


• A constant term, \(-1\), which stands alone without any variable attached.

Understanding the structure of a trinomial helps in identifying methods such as the AC method for factoring.

AC method

The AC method is a systematic approach used to factor trinomials, especially when simple factoring isn't evident. This method is effective when dealing with a trinomial in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In the exercise, we have \(a = 35\), \(b = -2\), and \(c = -1\). To use the AC method, follow these steps:

• Multiply \(a\) and \(c\), which gives us the product \(AC = 35 \times (-1) = -35\).


• Identify two numbers that multiply to \(-35\) and add to \(-2\). Here, the numbers are \(-7\) and \(5\).


• Rewrite the middle term \(-2w\) using the identified numbers: \(35w^2 -7w + 5w -1\).


• Form groups: \( (35w^2 -7w) + (5w - 1) \).


• Factor out the greatest common factor in each group: \(7w(5w - 1) + 1(5w - 1)\).


• Since both groups have a common binomial factor \((5w - 1)\), combine them: \((5w - 1)(7w + 1)\).

By carrying out the AC method precisely, we achieve a factorization that can easily be verified by expansion.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into multiplied factors that, when expanded, give the original polynomial. It is an essential skill in algebra that simplifies expressions and solves equations more efficiently. For the trinomial \(35w^2 - 2w - 1\), the factoring process helps in transforming it into \((5w - 1)(7w + 1)\).This process consists of several strategic steps:

• Analyze the polynomial to check if special patterns or simple common factors exist.


• If not, apply more advanced techniques like the AC method for trinomials.


• Once factored, multiply the factors to ensure they condense back to the original polynomial, confirming the accuracy of the factorization.

Factoring makes solving and interpreting polynomial equations easier, providing insights into the roots and behavior of polynomial functions.