Problem 110 - College Algebra

Question

In Exercises \(109-112,\) factor completely. $$-x^{2}-4 x+5$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(-x^{2} - 4x + 5\) is \(-(x-1)(x+5)\).

Step 1: Identify the Coefficients of the Polynomial

The given polynomial is \(-x^{2}-4x+5\). Here, the coefficient of \(x^{2}\) is -1, the coefficient of \(x\) is -4, and the constant term is 5.

Step 2: Find Two Numbers that Multiply to Constant Term and Add to Linear Coefficient

We need to find two numbers that multiply to 5 (the constant term), and add up to -4 (the coefficient of the linear term). The numbers that satisfy these conditions are -5 and 1, since \(-5*1=5\), and \(-5+1=-4\).

Step 3: Factor the Polynomial

Using these two numbers, we factor the polynomial as follows: \(-x^{2} - 4x + 5 = -(x-1)(x+5)\).

Key Concepts

Polynomial Coefficients

Polynomial coefficients are the numbers that appear before the variables in a polynomial expression and they determine the weight of each term in the polynomial. They are critical when it comes to understanding the behavior of the polynomial. For example, the polynomial in the exercise is \( -x^{2}-4x+5 \). In this expression, the coefficients are the numbers -1, -4, and 5. The coefficient of \( x^{2} \) especially, which is -1 in this case, dictates the polynomial's curvature, indicating that the graph opens downwards since it is negative. Recognizing these coefficients is the first step in the process of factoring polynomials.

When faced with a polynomial expression, students should practice identifying the coefficients as they directly influence the shape and position of the graph of the polynomial function. This awareness makes it easier to visualize the potential factors and eventual solutions of the equation.

Constant Term

The constant term in a polynomial expression is the value without any variable attached to it. In essence, it's the term's coefficient that stands alone. Referring to the provided polynomial \( -x^{2}-4x+5 \), the constant term is 5. This term is particularly important in factoring because it's a key component in finding potential factors of the polynomial. When factoring, you'll look for two numbers that multiply together to make the constant term.

In our exercise, to find numbers that when multiplied give us the constant term (5), we think of number pairs like (1, 5) or (-1, -5), always mindful that the constant can sometimes be negative as well, which would change our pairs accordingly. This is integral knowledge when attempting to split the middle term or when applying techniques such as the 'factor by grouping' method.

Linear Coefficient

The linear coefficient is the coefficient of the term with the variable raised to the first power, or simply put, the term \( x \). In the given polynomial \( -x^{2}-4x+5 \), the linear coefficient is -4. This coefficient plays a pivotal role when we split the middle term in factoring, as we seek two numbers that not only multiply to the constant term but also add up to the linear coefficient.

In the solution process of our example, the numbers -5 and 1, when added together, equal -4 (the linear coefficient), and their product is 5 (our constant term). Recognizing this relationship is a cornerstone in solving the factoring problem correctly, and it is often the trickiest part of factoring polynomials.