Problem 18 - Algebra and Trigonometry

Question

\(13-20\) . Factor the trinomial. $$ 5 x^{2}-7 x-6 $$

Step-by-Step Solution

Answer

The factorization of the trinomial is \((5x + 3)(x - 2)\).

Step 1: Identify a, b, and c in the Quadratic Expression

In the given quadratic trinomial, identify the coefficients where \( a = 5 \), \( b = -7 \), and \( c = -6 \). These correspond to the general form \( ax^2 + bx + c \).

Step 2: Determine Two Numbers that Multiply to ac and Add to b

Calculate \( ac = 5 \times -6 = -30 \). We need to find two numbers that multiply to -30 and add up to -7. These numbers are -10 and 3, since \(-10 \times 3 = -30\) and \(-10 + 3 = -7\).

Step 3: Rewrite the Middle Term Using the Two Numbers

Rewrite \(-7x\) as \(-10x + 3x\). The expression becomes \(5x^2 - 10x + 3x - 6\).

Step 4: Factor by Grouping

Group the terms into pairs: \((5x^2 - 10x) + (3x - 6)\). Factor out the greatest common factor from each pair. This gives us \(5x(x - 2) + 3(x - 2)\).

Step 5: Factor Out the Common Binomial

Observe that \((x - 2)\) is common in both terms. Factor \((x - 2)\) out, resulting in \((5x + 3)(x - 2)\).

Step 6: Verify the Factorization

Expand \((5x + 3)(x - 2)\) to verify. Performing the multiplication: \(5x \cdot x + 5x \cdot (-2) + 3 \cdot x + 3 \cdot (-2) = 5x^2 - 10x + 3x - 6\), which simplifies back to the original expression \(5x^2 - 7x - 6\).

Key Concepts

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree 2, usually expressed as \( ax^2 + bx + c \). This simply means that the highest exponent of the variable \( x \) is 2. Quadratic expressions often arise in various problems and are fundamental in algebra.

- **Why Quadratics Are Important:** - They frequently appear in nature and physics. - Used to describe parabolic paths such as projectile motion. - Allow for modeling a variety of real-world situations.
- **Components of a Quadratic Expression:** - **\( a \)**: The coefficient of \( x^2 \), which impacts the width and direction of the parabola. - **\( b \)**: The coefficient of \( x \), often influencing the parabola's symmetry. - **\( c \)**: The constant term, which sometimes determines where the graph intersects the y-axis.
Recognizing these components is crucial when working with quadratic expressions, as they help identify the properties and solutions related to the expression.

What is a Trinomial?

A trinomial is a specific type of polynomial with exactly three terms. In the context of quadratic expressions, a quadratic trinomial is a trinomial of the form \( ax^2 + bx + c \), where the degree of the trinomial is 2.

- **Characteristics of a Trinomial:** - Consists of three terms. - Usually arranged in descending powers of the variable. - In quadratic trinomials, terms include the squared term, linear term, and constant.
- **Importance in Algebra:** - Trinomials are common in solving quadratic equations. - They are a building block of various algebraic manipulations. - Understanding trinomials is key to mastering factorization and solving equations efficiently.
In our exercise, \( 5x^2 - 7x - 6 \) is an example of a quadratic trinomial. Recognizing it as such helps in utilizing methods like factoring by grouping to find its factors.

Factoring by Grouping Method

Factoring by grouping is a method used to factor polynomials, including trinomials like \( 5x^2 - 7x - 6 \). This technique involves dividing the polynomial into groups to simplify and factor each part step by step.

- **Steps to Factor by Grouping:** - **Split the Middle Term:** Start by finding two numbers whose product is \( ac \) and that add up to \( b \). These numbers will help split the middle term. - **Group the Terms:** After splitting, rearrange the trinomial as two binomial terms. This makes it easier to factor each group. - **Find Common Factors:** In each group, factor out the greatest common factor (GCF). - **Factor out Common Binomial:** Look for a common binomial in your grouped expression and factor it out.
- **Effective Usage:** - Is especially helpful for factoring quadratic trinomials that don't factor easily. - Allows us to transform complex expressions into simpler, more useful forms.
For the exercise given, after identifying \( a = 5 \), \( b = -7 \), and \( c = -6 \), we split \(-7x\) into \(-10x + 3x\), group into \((5x^2 - 10x) + (3x - 6)\), and factor each to reach \((5x + 3)(x - 2)\). This demonstrates how grouping can efficiently solve and factorize quadratic trinomials.