Problem 22 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Check your answer. $$s^{2}+3 s-28$$

Step-by-Step Solution

Answer

The completely factored form of the quadratic expression \(s^2 + 3s - 28\) is \((s + 4)(s - 7)\).

Step 1: Identify the quadratic expression

We have a quadratic expression in the form \(ax^2 + bx + c\), specifically: \[s^2 + 3s - 28\]

Step 2: Determine the leading coefficient, middle term, and constant

In this case, the leading coefficient (a) is 1, the middle term (b) is 3, and the constant (c) is -28.

Step 3: Find two numbers that multiply to the constant and add to the middle term

We need to find two numbers that multiply to -28 and add to 3. We can list the factors of -28: 1 and -28 || -1 and 28 2 and -14 || -2 and 14 4 and -7 || -4 and 7 From these factors, we can see that \(4 * -7 = -28\) and \(4 + -7 = 3\), which are our desired values.

Step 4: Write the factored form

Using the two numbers found in step 3, we write the factored form of the quadratic expression: \((s + 4)(s - 7)\]

Step 5: Check the answer

Now we simply expand the factored form and compare it to the original expression to check our answer: \((s + 4)(s - 7) = s^2 - 7s + 4s - 28 = s^2 + 3s - 28\) Since the expanded form matches the original expression, our factored form is correct: \[s^2 + 3s - 28 = (s + 4)(s - 7)\]

Key Concepts

Quadratic Expressions

Quadratic expressions are mathematical expressions where the variable is raised to the power of two. These expressions typically take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. The highest degree of the variable is what classifies it as a quadratic expression. For example, in the expression \(s^2 + 3s - 28\), \(s^2\) is the quadratic term, \(3s\) is the linear term, and \(-28\) is the constant term.

Understanding the structure of a quadratic expression helps in factoring it, which means breaking it down into simpler expressions that multiply to form the original expression. This is crucial in solving quadratic equations and simplifying mathematical problems that involve quadratics.

Factored Form

Factored form is the representation of a polynomial as a product of its factors. For quadratic expressions, factoring is a method used to simplify the expression into two binomial expressions. For the quadratic \(s^2 + 3s - 28\), the factored form is \((s + 4)(s - 7)\).

To achieve the factored form, one must find two numbers that multiply to the constant term and add to the middle coefficient (from the linear term). In this case, the numbers are 4 and -7.

Factoring is a useful tool in simplifying equations and solving for roots of quadratic expressions. It also helps in graphing quadratics by providing insight into the symmetry and zero points of the parabolic graph.

Leading Coefficient

The leading coefficient in a quadratic expression is the coefficient of the term with the highest power. In a standard quadratic form \(ax^2 + bx + c\), \(a\) is the leading coefficient. It is crucial because it affects the orientation and width of the parabola when graphed.

For the expression \(s^2 + 3s - 28\), the leading coefficient is 1. A leading coefficient of 1 means the parabola is neither compressed nor stretched compared to the standard parabola \(y = x^2\). If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.

Constant Term

The constant term in a quadratic expression is the term that does not include the variable. In the expression \(ax^2 + bx + c\), \(c\) is the constant term. This term is important in factoring since it represents the product of the numbers required for the factored form of the quadratic.

In our example, \(s^2 + 3s - 28\), the constant term is \(-28\). This value must be considered when finding two numbers that will allow for the expression to be factored into binomials, as they must multiply to \(-28\) and add to the middle coefficient (3 in this case). Understanding the role of the constant term aids in visualizing the vertical shift of the graph of the quadratic expression.