Problem 18 - Elementary and Intermediate Algebra

Question

Complete each step of the factorization of the trinomial by grouping. $$ \begin{aligned} 35 t^{2}-11 t-6 &=35 t^{2}+10 t-21 t-6 \\ &=5 t(7 t+2) \\_ 3(7 t+2) \\ &=(\quad)(5 t-3) \end{aligned} $$

Step-by-Step Solution

Answer

The trinomial factors into \((7t + 2)(5t - 3)\).

Step 1: Rearrange and Group Terms

First, we need to rearrange the middle terms of the trinomial. The trinomial is given as \(35t^2 - 11t - 6\). We look for two numbers whose product is \(35 \times (-6) = -210\) and whose sum is \(-11\). Those numbers are \(10\) and \(-21\). So, we split and rearrange the middle term \(-11t\) into \(10t\) and \(-21t\). This gives us the expression: \(35t^2 + 10t - 21t - 6\).

Step 2: Group Terms

Next, group the terms into two pairs. The expression rearranges to \((35t^2 + 10t) + (-21t - 6)\). We will factor each pair separately.

Step 3: Factor Each Group

Look at the first group: \(35t^2 + 10t\). The greatest common factor (GCF) is \(5t\), so we factor it out: \(5t(7t + 2)\). Now for the second group: \(-21t - 6\), the GCF is \(-3\), so we factor it out: \(-3(7t + 2)\).

Step 4: Factor by Grouping

The expression now is \(5t(7t + 2) - 3(7t + 2)\). Notice the common factor \((7t + 2)\). Factor this common term out to get \((7t + 2)(5t - 3)\).

Step 5: Final Expression

The factored form of the trinomial \(35t^2 - 11t - 6\) is \((7t + 2)(5t - 3)\).

Key Concepts

Grouping in Algebra

Grouping in algebra is a useful technique to simplify expressions, particularly when factoring trinomials. The goal of grouping is to organize expressions in a way that makes it easier to identify common factors. By rearranging and grouping terms in a polynomial, we can factor each group separately, making the process more manageable.
For example, consider the trinomial: \(35t^2 - 11t - 6\). In this case, we first need to split the middle term \(-11t\) into two separate terms that simplify the factorization process. We find numbers that multiply to the product of the first and last coefficients (here, \(35 \times -6 = -210\)) and add up to \(-11\). These numbers are \(10\) and \(-21\).

• Rearrange the trinomial as \(35t^2 + 10t - 21t - 6\).
• Group the terms: \((35t^2 + 10t) + (-21t - 6)\).

This strategic arrangement allows for term-by-term factorization, which is crucial in polynomial factorization.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by two or more expressions. It is a key concept in simplifying algebraic expressions and plays an important role when factoring polynomials, as it helps to reduce each part of the expression to its simplest form.
To factor the trinomial \(35t^2 - 11t - 6\) by grouping, we first identify the GCF of each pair of grouped terms.

• For the first pair, \(35t^2 + 10t\), the GCF is \(5t\). Factoring out \(5t\), we get \(5t(7t + 2)\).
• For the second pair, \(-21t - 6\), the GCF is \(-3\). Factoring out \(-3\), we get \(-3(7t + 2)\).

By identifying and factoring out the GCF in each group, we can simplify and solve complex expressions efficiently. This step-by-step approach is a cornerstone in algebra and essential for mastering polynomial factorization.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. It simplifies complex expressions and is useful for solving equations and understanding polynomial behavior.
To factorize the trinomial \(35t^2 - 11t - 6\), we use grouping, making it easier to factor each part by finding the common factors.

• After grouping and factoring out the GCF from each group (\(5t\) from the first group and \(-3\) from the second), we obtain the expression \(5t(7t + 2) - 3(7t + 2)\).
• Notice the common factor \((7t + 2)\) in both terms.
• Factor out the common term to arrive at the final factored form: \((7t + 2)(5t - 3)\).

By transforming polynomials through factorization, we can greatly simplify the problem-solving process in algebra, making it a fundamental skill for students to master.