Problem 37 - Algebra 1: Concepts and Skills

Question

Factor the trinomial. $$ 6 t^{2}+t-70 $$

Step-by-Step Solution

Answer

The factored form of the given trinomial \(6 t^{2}+t-70\) is \((2t - 7)(3t + 10)\).

Step 1: Identify the coefficients

Identify the coefficients of the trinomial where \(a = 6\), \(b = 1\), and \(c = -70\) which corresponds to the general form \(ax^2+bx+c\).

Step 2: Factor the product ac

Find two numbers whose product is \(ac = -420\) (which is the product of \(a\) and \(c\)) and their sum is \(b = 1\). The numbers are 20 and -21, as \(20 × -21 = -420\) and \(20 - 21 = -1\).

Step 3: Rewrite the trinomial

Rewrite the original equation replacing \(t\) with the pair found in step 2 as such: \(6t^{2} + 20t - 21t - 70 = 0\).

Step 4: Factor by grouping

Group the terms two at a time to perform factoring, acquiring \(2t(3t+10) -7(3t+10)\).

Step 5: More factorization

Here, we can see that \(3t+10\) is a common factor. So, the trinomial factors to \((2t - 7)(3t + 10)\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \). These equations can describe different types of problems, including those involving motion, areas, and other real-world phenomena. In the quadratic expression, \(a\), \(b\), and \(c\) are coefficients. Each term holds a vital role: \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term.

Quadratic equations can be solved by various methods, such as:

• Factoring
• Completing the square
• Using the quadratic formula

Factoring is often the simplest and quickest method, especially with manageable numbers. However, it's essential to identify when a quadratic can be factored easily. For instance, the equation \(6t^2 + t - 70 = 0\) can be broken down into simpler linear factors. Once it is factored, the solutions to the equation can be found by setting each factor equal to zero. This process helps in finding the values of \(x\) that satisfy the original equation.

Polynomial Factorization

Factoring polynomials is breaking down a complex polynomial into simpler products. This technique makes handling large expressions more manageable and reduces the complexity when solving equations. With trinomials like \(6t^2 + t - 70\), polynomial factorization plays a crucial role.

To begin with factoring polynomial expressions, you need to:

• Identify the coefficient values \(a\), \(b\), and \(c\)
• Calculate the product \(ac\)

Look for two numbers whose product equals \(ac\) and their sum equals \(b\). This approach is the heart of the method called factoring by grouping. Factoring by grouping involves rearranging and splitting the middle term of the trinomial into two terms whose coefficients add up to \(b\). The next steps, such as forming groups and identifying common factors, help in breaking the trinomial into a product of binomials. The beauty of polynomial factorization becomes evident as you streamline what seems complex into something easily manageable.

Basics of Algebraic Expressions

Algebraic expressions form the foundation of algebra and can encompass variables, numbers, and operations. For example, expressions like \(6t^2 + t - 70\) carry significant importance because they represent more intricate algebraic rules and concepts.

To delve deeper into algebraic expressions, it's helpful to understand:

• What variables and coefficients are
• The operations involved, such as addition, subtraction, and multiplication
• The structure of polynomials and how they can be simplified or rearranged

The core of handling algebraic expressions resides in recognizing their forms and manipulating them to solve for unknown values. Factoring is one of these manipulation techniques, particularly useful for solving equations and simplifying expressions. It transforms a broad, unwieldy expression into a set of multiplied expressions, making the solving process manageable and less error-prone. As you master simplifying these expressions, you open doors and opportunities for understanding more advanced algebraic concepts with confidence.