Problem 75 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$4 n^{2}+25 n+36$$

Step-by-Step Solution

Answer

The polynomial factors completely as \((4n + 9)(n + 4)\) using integers.

Step 1: Identify the Polynomial Type

The given polynomial is a quadratic polynomial in the form of \(4n^2 + 25n + 36\). We need to factor it using techniques such as grouping or the quadratic formula if needed.

Step 2: Check for a Common Factor

Examine all the terms, \(4n^2\), \(25n\), and \(36\), to see whether there is a common factor. Since there is no common factor other than 1, we proceed to the next step.

Step 3: Apply the Quadratic Formula

The polynomial is quadratic, so we can use the quadratic formula to find the roots if needed. The formula is \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \(a = 4\), \(b = 25\), and \(c = 36\).

Step 4: Calculate the Discriminant

Compute the discriminant \(b^2 - 4ac\). Here, \(b = 25\), \(a = 4\), \(c = 36\), so calculate: \(25^2 - 4 \times 4 \times 36 = 625 - 576 = 49\).

Step 5: Apply the Square Root of the Discriminant

Since the discriminant is a perfect square (\(49\)), we can factor the quadratic polynomial into two binomials. Calculate the square root: \(\sqrt{49} = 7\).

Step 6: Find the Factors

Using the roots from the quadratic equation, \(n = \frac{-25 \pm 7}{8}\). The solutions are \(n = \frac{-18}{8} = -\frac{9}{4}\) and \(n = \frac{-32}{8} = -4\). Thus, the polynomial factors to \((n + \frac{9}{4})(n + 4)\).

Step 7: Adjust Factor to Integer Coefficients

Since we need integer coefficients, multiply both factors by 4 to clear the fraction: \((4n + 9)(n + 4)\). So, the polynomial factors completely as \((4n + 9)(n + 4)\).

Key Concepts

Quadratic Formula

The quadratic formula is a key tool used in algebra to solve quadratic equations, which have the form \(ax^2 + bx + c = 0\). This formula provides the roots of the equation, which are the values of \(x\) for which the equation equals zero. The quadratic formula is given by:\[n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• a, b, c: These are constants from the quadratic equation.
• \(\pm\): Indicates that there are generally two solutions: one using the plus sign, and one using the minus sign.

Understanding how to apply the quadratic formula involves plugging in the coefficients \(a\), \(b\), and \(c\) from your equation into the formula, and solving for \(x\). For instance, in the exercise above, the coefficients are \(a = 4\), \(b = 25\), \(c = 36\), allowing us to solve for \(n\). This provides valuable insights into the factorization process.

Discriminant

The discriminant is an essential part of the quadratic formula and plays a pivotal role in determining the nature of the roots of a quadratic equation. The discriminant is the part of the quadratic formula under the square root, calculated as \(b^2 - 4ac\).

• Positive Discriminant: If the discriminant is positive, the quadratic equation has two distinct real roots. For example, in our exercise, \(b^2 - 4ac = 49\), which is positive.
• Zero Discriminant: If the discriminant is zero, there is exactly one real root, meaning the quadratic touches the x-axis at one point.
• Negative Discriminant: A negative discriminant indicates that the quadratic has no real roots and the roots are complex numbers.

In this exercise, since the discriminant (49) is a perfect square, it suggests that the quadratic can be factored into binomials with integer coefficients, aiding in the polynomial's complete factorization.

Common Factors

Before diving into more complex factorization techniques, evaluating a polynomial for any common factors is a critical initial step. A common factor is a number or expression that divides each term of the polynomial without leaving a remainder.

• Identifying Common Factors: Look at each term of the polynomial. For instance, in the quadratic \(4n^2 + 25n + 36\), the terms are checked for factors. Here, none exist besides 1.
• If Found: If there are common factors, you can factor them out to simplify further solving.

Understanding that this step simplifies the process and sometimes results in easier expressions to work with before applying other methods like the quadratic formula.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler "factor" forms that, when multiplied, give back the original polynomial. Factoring can sometimes be challenging but simplifies problem-solving considerably.

• Using Binomial Factors: Often involves expressing the polynomial as a product of binomials. For our equation, after using the quadratic formula and identifying the roots, we transformed \(4n^2 + 25n + 36\) into the binomials \((n + \frac{9}{4})(n + 4)\).
• Adjusting for Integer Coefficients: If rational factors aren't in integer form, you may need to adjust. In this case, multiplying by 4 results in the final integer factorization: \((4n + 9)(n + 4)\).

Factoring polynomials reduces them to manageable parts, aiding in finding solutions and gaining insights into their properties. This is especially useful in solving equations and understanding their behavior graphically.