Problem 38 - Algebra and Trigonometry

Question

Find all the real zeros of the function: $$ f(x)=(x-2)\left(x^{2}-3 x-10\right) $$

Step-by-Step Solution

Answer

The real zeros of the function are: \( x = 2, x = 5, x = -2 \).

Step 1 - Identify the Factors

The given function is factored into two parts: \( f(x) = (x-2)(x^2 - 3x - 10) \). First, identify the factored parts: \( x-2 \) and \( x^2 - 3x - 10 \).

Step 2 - Set Each Factor to Zero

To find the zeros of the function, set each factor equal to zero. This means solving the equations:\[ x - 2 = 0 \] and \[ x^2 - 3x - 10 = 0 \].

Step 3 - Solve for the First Factor

Solve the first equation: \( x - 2 = 0 \). This gives: \[ x = 2 \].

Step 4 - Solve for the Second Factor

Solve the quadratic equation: \( x^2 - 3x - 10 = 0 \). Factorize the quadratic: \( x^2 - 3x - 10 = (x - 5)(x + 2) \).

Step 5 - Find the Zeros of the Quadratic

Set each factor of the quadratic to zero:\[ x - 5 = 0 \] and \[ x + 2 = 0 \]. This gives: \[ x = 5 \] and \[ x = -2 \].

Step 6 - Compile All Zeros

Combine all the solutions to identify all the zeros of \( f(x) \). The zeros are: \[ x = 2, x = 5, x = -2 \].

Key Concepts

factoring polynomials

To find the zeros of a polynomial function, you often need to factor the polynomial. Factoring splits the polynomial into simpler pieces that are easier to handle.
In our given function, we have: \( f(x) = (x-2)(x^2 - 3x - 10) \)
Here, the polynomial \( f(x) \) is already partly factored with one factor being \( (x-2) \) and the other, a quadratic polynomial \( (x^2 - 3x - 10) \).
For quadratic terms, you look for two numbers that multiply to the constant term and add to the coefficient of the middle term. In this case, we need to find two numbers that multiply to -10 (the constant term) and add to -3 (the coefficient of \( x \)).
Those numbers are -5 and 2. Hence, the quadratic polynomial \( x^2 - 3x - 10 \) factors into \( (x-5)(x+2) \).
Factoring simplifies solving the polynomial equation, making it easier to find its zeros. Keep practicing identifying factor pairs, especially with different types of polynomials (quadratics, trinomials, etc.).

solving quadratic equations

Once the polynomial is factored, you can solve for the zeros. For quadratic equations, which are polynomials of degree 2, there are a few common methods to find solutions:

• Factoring: Splitting the equation into simpler binomials as we did with \( x^2 - 3x - 10 = (x-5)(x+2) \).


• Quadratic Formula: If factoring is too challenging, you can always use the quadratic formula:
  \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  For \( ax^2 + bx + c = 0 \), simply plug in the coefficients \( a \), \( b \), and \( c \) and solve for \( x \).


• Completing the Square: This involves converting the quadratic into perfect square form and then solving. It's more advanced but can be useful in some cases.

In our example, factoring was straightforward:
\( x^2 - 3x - 10 = (x-5)(x+2) \)
Solve each binomial accordingly: \( x-5 = 0 \) yields \( x = 5 \) and \( x+2 = 0 \) yields \( x = -2 \).
So, the solutions are x = 5 and x = -2.
Using these methods, you can solve any quadratic equation efficiently.

finding zeros of polynomial functions

The main goal with polynomials is often finding their zeros, the points at which they cross the x-axis. These zeros are the solutions to the equation when the function equals zero.
To find these zeros:

• Step 1: Factor the polynomial as much as possible. For our example, \( f(x) = (x-2)(x^2 - 3x - 10)\), we factored further to \( f(x) = (x-2)(x-5)(x+2) \).


• Step 2: Set each factor equal to zero and solve. This means solving \( x-2 = 0 \), \( x-5 = 0 \), and \( x+2 = 0 \).


• Step 3: Combine your solutions. Here, the solutions are x = 2, x = 5, and x = -2.

By following these steps, you can pinpoint all the points where your polynomial equation equals zero.
Remember, zeroes of a function imply the x-values where the function touches or crosses the x-axis.
Simplifying and understanding each polynomial, step-by-step, is crucial for mastering these concepts. Keep practicing with different examples to become more familiar with the process!