Problem 17 - Algebra and Trigonometry

Question

\(13-20\) . Factor the trinomial. $$ 3 x^{2}-16 x+5 $$

Step-by-Step Solution

Answer

The factors of the trinomial are \((3x - 1)(x - 5)\).

Step 1: Identify and Write Down the Trinomial

The trinomial given is \(3x^2 - 16x + 5\). We will factor this trinomial into the product of two binomials.

Step 2: Determine the Product and Sum

To factor the trinomial \(3x^2 - 16x + 5\), we need two numbers that multiply to \(3 \times 5 = 15\) and add to \(-16\).

Step 3: Find Two Numbers

The two numbers that multiply to 15 and add to -16 are -15 and -1. These numbers will help us split the middle term.

Step 4: Rewrite the Middle Term

Rewrite \(-16x\) as \(-15x - x\) using the numbers found: \(3x^2 - 15x - x + 5\).

Step 5: Group the Terms

Group the terms: \((3x^2 - 15x) - (x - 5)\).

Step 6: Factor Each Group

Factor out the greatest common factor from each group: \(3x(x - 5) - 1(x - 5)\).

Step 7: Factor by Grouping

Notice \((x - 5)\) is a common factor: \((3x - 1)(x - 5)\).

Step 8: Verify the Factors

Check by expanding \((3x - 1)(x - 5)\) to ensure it equals the original trinomial \(3x^2 - 16x + 5\).

Key Concepts

Polynomial Decomposition

Polynomial decomposition involves breaking a polynomial, such as a trinomial, down into simpler components. This is often in the form of products (or factors) that, when multiplied together, give back the original polynomial.

In the case of trinomials, particularly of the form \(ax^2 + bx + c\), decomposition helps in identifying which binomial factors, when multiplied, will reconstruct the original expression. The goal is to express the trinomial as the product of two binomials.

• Start by identifying the trinomial you need to factor. For our example, it is \(3x^2 - 16x + 5\).
• The next step is to find two numbers whose product is the first coefficient times the last coefficient (in this case, \(3 \times 5 = 15\)), and whose sum is the middle coefficient (-16).
• With these numbers (-15 and -1), the polynomial can be rewritten, aiding the decomposition.

This method is particularly helpful in simplifying expressions, making it easier to solve or manipulate algebraic equations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can be as simple as a single variable or more complex, involving multiple terms and operations like addition, subtraction, multiplication, and division.

In the trinomial \(3x^2 - 16x + 5\), each part of the expression has a role:

• \(3x^2\): The term with \(x^2\) indicates this is a quadratic expression.
• \(-16x\): This linear term has a coefficient of -16, which affects how the expression behaves when graphed.
• \(+5\): The constant term shifts the graph of the expression vertically.

These expressions are essential in algebra because they help to model real-world situations, solve equations, and understand relationships between quantities. Factoring simplifies these expressions, making them more manageable and revealing their underlying structure.

Quadratic Equations

Quadratic equations are a key area of study in algebra and often take the form \(ax^2 + bx + c = 0\). These equations can be solved by various methods, including factoring, completing the square, or using the quadratic formula. The goal is to find the values of \(x\) for which the equation is true, known as the solutions or roots.

Understanding the quadratic trinomial, like \(3x^2 - 16x + 5\), is crucial for factoring it effectively. Each component of this expression relates directly to its behavior when set equal to zero in a quadratic equation.

• Factoring transforms the equation into \((3x - 1)(x - 5) = 0\), simplifying the process of finding roots.
• Setting each factor equal to zero gives \(3x - 1 = 0\) and \(x - 5 = 0\).
• Solving these, we find \(x = \frac{1}{3}\) and \(x = 5\) are the roots.

This method highlights the importance of factoring when solving quadratic equations, providing a straightforward path to unveiling their solutions.