Problem 67 - Algebra 2

Question

State whether each trinomial is a perfect square. If so, factor it. \(x^{2}-5 x-10\)

Step-by-Step Solution

Answer

The trinomial is not a perfect square.

Step 1: Understand Perfect Square Trinomials

A perfect square trinomial is one that can be expressed as \( (a-b)^2 = a^2 - 2ab + b^2 \) or \( (a+b)^2 = a^2 + 2ab + b^2 \). Let's identify if the given trinomial follows this form.

Step 2: Identify 'a' and 'b' in the Trinomial

The standard form suggests that you should compare the given trinomial with the perfect square form \( a^2 \, + \, 2ab \, + \, b^2 \) or \( a^2 \, - \, 2ab \, + \, b^2 \).The first term, \( x^2 \), suggests that \( a = x \).

Step 3: Determine Possible 'b' and Compare

The original expression is \( x^2 - 5x - 10 \).To be a perfect square trinomial, \( -10 \) would need to be \( b^2 \), and \( -5x \) would need to be \( -2ab \).Let's compute and verify these conditions.

Step 4: Check the Middle Term

If the trinomial is a perfect square, then \(-2ab = -5x\).With \(a = x\), that gives \(-2b = -5\), hence \(b = \frac{5}{2}\).

Step 5: Verify the Constant Term

If \( b = \frac{5}{2} \), then \( b^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} \).However, \(-10\) is not equal to \(\frac{25}{4}\), indicating that the trinomial is not a perfect square.

Key Concepts

Factoring Trinomials

Factoring trinomials is a key process in algebra that involves breaking down a quadratic expression into the product of two binomials. This method is used to simplify expressions and solve quadratic equations more easily.
One of the well-known forms that trinomials can take is the **perfect square trinomial**. Understanding this specific form helps identify when a trinomial can be neatly factored.
When looking to factor a trinomial like \(x^2 - 5x - 10\), it is crucial to determine whether it fits into the perfect square form or not. A perfect square trinomial can be expressed in the form \((a + b)^2\) or \((a - b)^2\), which simplifies to \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\) respectively.

• First, identify if the trinomial meets the conditions of a perfect square.
• Check if the first term is a square and if the last term is a square.
• Determine if the middle term is twice the product of their respective square roots.

For our trinomial, \(x^2 - 5x - 10\), identifying the perfect square conditions verifies whether standard factoring methods apply or more advanced techniques are needed.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two, typically written in the standard form of \(ax^2 + bx + c\). Such expressions often appear in algebra problems and require specific techniques for simplification and solving.
Understanding the structure of quadratic expressions is essential for recognizing various patterns, such as perfect squares or factorable forms. Remember:

• The term \(ax^2\) represents the leading term that defines the degree of the polynomial.
• The term \(bx\) determines the linear component of the expression.
• The constant term \(c\) has a significant role when identifying certain patterns like perfect squares.

When evaluating \(x^2 - 5x - 10\), it's important to understand how each part of the expression contributes to its factorization. For example, the middle term, \(-5x\), informs us about potential factors when compared to the product of terms in potential binomials. Focusing on these components allows us to discern whether specific factoring techniques, such as simple factorization or completing the square, are applicable.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They serve as useful tools when simplifying expressions and solving equations in algebra. Some common algebraic identities you might encounter include:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)
• \(a^2 - b^2 = (a + b)(a - b)\)

These identities assist in recognizing or creating certain forms within algebraic expressions. The **perfect square trinomial** is a great example of using algebraic identities.
In our given problem, we examine \(x^2 - 5x - 10\) to verify if it matches a perfect square identity. Recognizing these patterns requires comparison of terms and realizing if and how they can fit into the identity's structure. If they do not align, it indicates that the trinomial is not a perfect square and we might need to explore different approaches for further simplification or solving.