Problem 67 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Begin by asking yourself, "Can I factor out a GCF?" $$s^{2}+4 s t+5 t^{2}$$

Step-by-Step Solution

Answer

The given expression \(s^{2}+4st+5t^{2}\) cannot be factored further as there are no common factors among the terms and the trinomial doesn't satisfy the conditions for factoring into binomials. Therefore, the final answer is: \(s^{2} + 4st + 5t^{2}\)

Step 1: Identify the GCF of the terms.

The given polynomial is \(s^{2}+4st+5t^{2}\). Let's check if there is any greatest common factor (GCF) among the terms. For the first term, the factors are \(s * s\). For the second term, the factors are \(s * t * 2 * 2\). For the third term, the factors are \(t * t * 5\). As there is no common factor among these three terms, we cannot factor out a GCF. Step 2: Factor by grouping or other techniques

Step 2: Try factoring the trinomial.

Since there is no GCF to factor, we move on to try factoring the trinomial. The given expression is in the form \(s^{2}+4st+5t^{2}\), which resembles a quadratic equation in the form \(ax^2 + bx + c\), where the variable x is replaced by st. This suggests that we might be able to factor the expression by looking for two binomials that multiply to equal the original expression. In this case, we want to find two binomials \((as + bt)\) and \((cs + dt)\) such that the product of these binomials is equal to \(s^{2}+4st+5t^{2}\). Let's expand the product of these binomials and match the coefficients: \((as + bt)(cs + dt) = s^{2}+4st+5t^{2}\) Expanding the product: \(ac s^2 + (ad + bc) st + bd t^2 = s^2 + 4st + 5t^2\) Matching coefficients, we have the following system of equations: 1. \(ac = 1\) 2. \(ad + bc = 4\) 3. \(bd = 5\) We notice that the only possible pairs of factors for 1 and 5 are (1, 1) and (5, 1) respectively. After trying out these combinations and solving for a, b, c, and d, we can observe there are no solutions that satisfy all three equations. Therefore, this trinomial cannot be factored further. The final answer is: \(s^{2} + 4s t + 5t^{2}\)

Key Concepts

Greatest Common Factor

When dealing with polynomials, identifying the greatest common factor (GCF) is usually the first step in simplifying or factoring them. The GCF is the largest factor that divides each term in a polynomial without leaving a remainder. To find the GCF:

• List the factors of each term in the polynomial.
• Identify the common factors shared by all terms.
• Choose the highest factor common to each term.

In the original problem, the polynomial is given as \(s^{2}+4st+5t^{2}\). The factors of each term are checked for commonality, but no GCF is found because the three terms \(s^{2}\), \(4st\), and \(5t^{2}\) share no common factor. Hence, we move on to other factoring techniques. Recognizing when no GCF is present helps direct efforts towards alternative strategies for simplifying the polynomial.

Trinomial Factoring

Trinomial factoring involves expressing a trinomial as a product of two binomials. A trinomial typically takes the form \(ax^2 + bx + c\). The method of factoring depends on identifying two numbers that multiply to \(ac\) while adding to \(b\).
With the expression \(s^2 + 4st + 5t^2\), we attempt to write it as \((as + bt)(cs + dt)\). To do this:

• Expand the product \((as + bt)(cs + dt)\) to reproduce the original trinomial.
• Compare the coefficients from the expanded form to those in the original trinomial.

This involves testing different combinations of factors of the constant term. In this case, the factors of 5 and 1 were explored, but no suitable combination was found. Thus, not all trinomials are factorable into integers, sometimes resulting in irreducible expressions.

Quadratic Equation

A quadratic equation is a polynomial of degree two that typically takes the form \(ax^2 + bx + c = 0\). Solutions to quadratic equations can be sought through various methods such as:

• Factoring, if possible.
• Completing the square.
• Applying the quadratic formula.

In the exercise, we're working with an expression \(s^2 + 4st + 5t^2\) that resembles a quadratic form. However, without attainable binomials corresponding to the necessary factors, traditional factoring methods do not simplify the trinomial further.
Understanding the limitations of factoring can aid in determining when to apply alternative methods for solving or simplifying quadratic expressions. Knowing these techniques provides a toolkit for tackling a wide range of quadratic problems.

Binomials

Binomials are algebraic expressions containing two terms. Commonly seen in forms like \(ax + b\), they are building blocks for constructing higher-degree polynomials through multiplication.

• Binomials are often used in factoring trinomials
• The form \((x + p)(x + q)\) is aimed at simplifying quadratic-like expressions
• Expansion of binomials uses the distributive property: \((x + p)(x + q) = x^2 + (p+q)x + pq\)

In this exercise, although the goal was to rewrite the trinomial \(s^2 + 4st + 5t^2\) using two binomials, no integer pairs \((as + bt)(cs + dt)\) satisfied the distribution and coefficient matching. While binomials are crucial in polynomial factoring, not every trinomial can be expressed as the product of binomials with integer components, emphasizing the importance of recognizing limitations during factor attempts.