Problem 111 - Introductory Algebra for College Students

Question

Find all integers \(b\) so that the trinomial can be factored. $$\text { Factor: } 3 x^{10}-4 x^{5}-15$$

Step-by-Step Solution

Answer

Given the trinomial, there is no such integer \(b\) that could be a multiplier for the \(x^5\) term in the trinomial. However, if \(b\) represents the power of \(x\), then the possible integer values are 3 and -5/3.

Step 1: Rewrite as quadratic trinomial

Rewrite the trinomial in terms of \(x^5\). We rewrite it as \(3(x^5)^2 -4x^5 -15 = 0\);

Step 2: Apply Quadratic Formula

Apply the quadratic formula to the rewritten equation. The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). In this equation, \(a=3\), \(b=-4\) and \(c=-15\). Apply these values to the quadratic formula:

Step 3: Calculate Discriminant

First, calculate the discriminant (the part under the square root in the quadratic formula), which is \(b^2 -4ac=(-4)^2 - 4*3*(-15)=16+180=196\). Since the discriminant is positive, there will be two distinct solutions.

Step 4: Calculate Quadratic Roots

Next, substitute the calculated discriminant back into the quadratic formula to calculate both roots: \(x = \frac{-(-4) \pm \sqrt{196}}{2*3} = \frac{4 \pm 14}{6}\). So the two roots are \(x = 3\) and \(x = -5/3\).

Step 5: Substitute Back

The equation was rewritten in terms of \(x^5\). So the solutions are \(x^5 = 3\) and \(x^5 = -5/3\). So the possible values of \(b\) are 3 and -5/3 if \(b\) represents \(x^5\). Otherwise, if \(b\) is the multiplier of \(x^5\) term, there are no such integer values for \(b\) in this trinomial as the coefficient of \(x^5\) is -4.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving equations of the form \(ax^2 + bx + c = 0\r\r\), where \(a\), \(b\), and \(c\) are known coefficients, and \(x\) represents the variable. This formula is given by \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\r\r\). It directly yields the roots of the quadratic equation, which are the values of \(x\) that satisfy the equation.

When applying the quadratic formula, the process involves identifying the coefficients \(a\), \(b\), and \(c\) from the quadratic equation and then substituting them into the formula. Afterward, simplifying the expression under the square root, known as the discriminant, and calculating the numerator and denominator leads to the solution for \(x\). The ± sign signifies that there will often be two solutions for \(x\), one for each sign.

Discriminant Calculation

The discriminant in a quadratic equation is the expression under the square root in the quadratic formula, \(b^2 - 4ac\). It is a critical part of the formula as it determines the nature and the number of roots the quadratic equation will have.

To calculate the discriminant:

• First, identify the coefficient \(b\) and square it.
• Then, multiply the coefficient \(a\) by the coefficient \(c\) and by 4.
• Subtract the product from the squared \(b\).

The value of the discriminant can tell us:

• If it is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (also called a repeated or double root).
• If it is negative, there are no real roots, but rather two complex roots.

The discriminant is not only a part of the process to find the roots but also gives insight into the characteristics of the equation without having to solve it.

Quadratic Roots

Quadratic roots are the solutions to the quadratic equation \(ax^2 + bx + c = 0\). For a quadratic equation in a standard form, roots can be found using the quadratic formula. Once the discriminant has been calculated, it is placed back into the formula to find the two potential values for \(x\).

The solutions or roots can be real or complex numbers. Real roots are the x-intercepts of the quadratic function on a graph when it is represented as \(y = ax^2 + bx + c\).

Real-life implications

Understanding quadratic roots is not only crucial for solving equations in algebraic settings but also has practical applications in fields like physics, engineering, and economics where models often involve quadratic equations.

Remember, when working with an equation that has been manipulated, such as from \(x^{10}\) to \(x^5\), it's imperative to relate the roots back to the original terms, which might involve taking the fifth root or any other operations to return to the original variable of the equation.