Problem 8 - Elementary and Intermediate Algebra

Question

Check to determine whether \((3 t-1)(5 t-6)\) is the correct factorization of \(15 t^{2}-19 t+6\).

Step-by-Step Solution

Answer

No, \((3t-1)(5t-6)\) is not the correct factorization of \(15t^2-19t+6\).

Step 1: Expand the Factored Expression

Let's expand the expression \((3t - 1)(5t - 6)\). Use the distributive property (also known as FOIL for binomials):- First, multiply the first terms: \(3t \times 5t = 15t^2\).- Outside, multiply the outer terms: \(3t \times -6 = -18t\).- Inside, multiply the inner terms: \(-1 \times 5t = -5t\).- Last, multiply the last terms: \(-1 \times -6 = 6\).

Step 2: Combine the Middle Terms

Now, let's combine the middle terms \(-18t\) and \(-5t\):\(-18t - 5t = -23t\).Now, collect all the results from the previous step together:\(15t^2 - 23t + 6\).

Step 3: Compare with the Original Expression

Now, compare the expanded expression,\(15t^2 - 23t + 6\), with the original polynomial, \(15t^2 - 19t + 6\).Notice that the two expressions differ in the coefficient of the linear term: \(-23 eq -19\).

Step 4: Conclusion

Since the expanded form\(15t^2 - 23t + 6\) does not match \(15t^2 - 19t + 6\), \((3t-1)(5t-6)\) is not the correct factorization of \(15t^2 - 19t + 6\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in algebra to simplify expressions and solve equations. It states that for any numbers or expressions, the multiplication of a sum by a number equals the sum of the individual products of each term within the parentheses with the number.
In mathematical terms, this can be represented as: \( a(b + c) = ab + ac \).

• First, multiply each term inside the parentheses by the number or term outside.
• Then, add up the resulting products.

In the context of the problem, we apply the distributive property to the binomials \((3t - 1)(5t - 6)\) using the FOIL method, which is a special case of the distributive property for binomials:

• First: Multiply the first terms of each binomial: \(3t \times 5t = 15t^2\).
• Outside: Multiply the outer terms: \(3t \times -6 = -18t\).
• Inside: Multiply the inner terms: \(-1 \times 5t = -5t\).
• Last: Multiply the last terms: \(-1 \times -6 = 6\).

By distributing each term, we can expand the original multiplication into a full polynomial.

Binomial Expansion

Binomial expansion involves expanding expressions that have two terms, or 'binomials,' when raised to a power or multiplied by another binomial. The process ensures that every possible combination of terms is accounted for.
Expanding binomials is crucial in factorization and solving polynomial equations. In this exercise, we expanded \((3t - 1)(5t - 6)\) using the distributive property:

• The product of the first terms gives us \(15t^2\).
• The outer and inner terms result in the additional coefficients of \(-18t\) and \(-5t\), respectively.
• Finally, the product from the last terms results in a constant of \(6\).

When we add up all these products, we get the expanded polynomial \(15t^2 - 23t + 6\).
Understanding the sequence of multiplication and combination is key when working with binomial expansion. Missteps in any part can lead to incorrect results, which highlights the importance of careful calculation.

Comparing Expressions

Comparing expressions is a vital skill in verifying factorization validity or identifying equivalent expressions. In our example, the purpose was to determine if \((3t - 1)(5t - 6)\) correctly factors into \(15t^2 - 19t + 6\).
This involved several steps:

• First, we expanded the binomial to see if it matches the original quadratic expression.
• After expanding, we found \(15t^2 - 23t + 6\).
• The important comparison is between the linear coefficients: the expanded form has \(-23t\) whereas the original has \(-19t\).

Because these coefficients do not match, we conclude that \((3t - 1)(5t - 6)\) doesn't factor \(15t^2 - 19t + 6\) correctly.
This process of comparing expressions ensures that our factorization or solving methods yield consistent results and help us spot possible errors or assumptions in calculations.