Finding Other Zeros Of Polynomial Functions

Hey guys! Today, we're diving deep into the world of polynomial functions and how to find their zeros. This is a crucial skill in algebra and calculus, and it's super useful for understanding the behavior of these functions. We'll tackle a specific example where we're given one zero and need to find the rest. So, buckle up and let's get started!

Understanding Polynomial Zeros

Polynomial zeros, also known as roots or x-intercepts, are the values of x that make the polynomial function equal to zero. In simpler terms, they are the points where the graph of the polynomial crosses or touches the x-axis. Finding these zeros is essential for various applications, such as solving equations, graphing functions, and modeling real-world phenomena.

When dealing with polynomial functions, it's important to remember the Fundamental Theorem of Algebra. This theorem states that a polynomial of degree n (the highest power of x in the polynomial) has exactly n complex roots, counting multiplicities. This means that a cubic polynomial (degree 3) will have three roots, a quartic polynomial (degree 4) will have four roots, and so on. These roots can be real numbers, imaginary numbers, or a combination of both. Real roots correspond to the x-intercepts of the graph, while imaginary roots do not appear on the graph.

Another key concept is the Factor Theorem. This theorem states that if a is a zero of a polynomial function f(x), then (x - a) is a factor of f(x). Conversely, if (x - a) is a factor of f(x), then a is a zero of f(x). This theorem provides a powerful tool for factoring polynomials and finding their zeros. For instance, if we know that 2 is a zero of a polynomial, then we know that (x - 2) is a factor of that polynomial. We can use this information to divide the polynomial and find the remaining factors and zeros.

The multiplicity of a zero refers to the number of times a particular zero appears as a root of the polynomial. For example, if (x - 2)² is a factor of a polynomial, then 2 is a zero with multiplicity 2. This means that the graph of the polynomial touches the x-axis at x = 2 but does not cross it. Zeros with odd multiplicities (like 1 or 3) will cause the graph to cross the x-axis, while zeros with even multiplicities (like 2 or 4) will cause the graph to touch the x-axis and turn around. Understanding the multiplicity of zeros is crucial for accurately sketching the graph of a polynomial function.

The Problem: Finding the Other Zeros

Okay, let's jump into our specific problem. We're given the polynomial function:

f(x) = x³ + 7x² - 5x - 35

And we know that x = -7 is one of the zeros. Our mission, should we choose to accept it (and we do!), is to find the other zeros.

Step 1: Using Synthetic Division

Since we know one zero, x = -7, we can use synthetic division to simplify the polynomial. Synthetic division is a nifty shortcut for dividing a polynomial by a linear factor like (x + 7). Here's how it works:

1. Set up: Write down the coefficients of the polynomial (1, 7, -5, -35) and the zero (-7) in the synthetic division format:

-7 | 1  7  -5  -35
   |________________

2. Bring down: Bring down the first coefficient (1) below the line:

-7 | 1  7  -5  -35
   |________________
    1

3. Multiply and add: Multiply the number you just brought down (1) by the zero (-7) and write the result (-7) under the next coefficient (7). Add these numbers (7 + (-7) = 0) and write the sum below the line:

-7 | 1  7  -5  -35
   |    -7
   |________________
    1  0

4. Repeat: Repeat the multiply and add process for the remaining coefficients. Multiply 0 by -7 (result is 0), write it under -5, and add (-5 + 0 = -5). Then, multiply -5 by -7 (result is 35), write it under -35, and add (-35 + 35 = 0):

-7 | 1  7  -5  -35
   |    -7   0   35
   |________________
    1  0  -5   0

The last number below the line (0) is the remainder. Since the remainder is 0, this confirms that x = -7 is indeed a zero of the polynomial. The other numbers below the line (1, 0, -5) are the coefficients of the quotient polynomial. In this case, the quotient is:

x² + 0x - 5 = x² - 5

Step 2: Solving the Quadratic Equation

Now we have a quadratic equation, x² - 5 = 0. We can solve this equation to find the remaining zeros. There are a couple of ways to do this:

• Factoring: In this case, factoring is straightforward. We can rewrite the equation as x² = 5. Taking the square root of both sides, we get:

  x = ±√5

• Quadratic Formula: If the quadratic equation is more complex and doesn't factor easily, we can use the quadratic formula:

  x = (-b ± √(b² - 4ac)) / (2a)

  In our case, a = 1, b = 0, and c = -5. Plugging these values into the quadratic formula, we get:

  x = (0 ± √(0² - 4 * 1 * -5)) / (2 * 1)

  x = (± √20) / 2

  x = (± 2√5) / 2

  x = ±√5

Both methods give us the same result: the other zeros are x = √5 and x = -√5.

The Answer

So, the other zeros of the polynomial function f(x) = x³ + 7x² - 5x - 35, given that -7 is a zero, are √5, -√5.

Key Takeaways

• Polynomial zeros are the values of x that make the function equal to zero.
• The Fundamental Theorem of Algebra tells us how many zeros a polynomial has.
• The Factor Theorem helps us relate zeros and factors of a polynomial.
• Synthetic division is a powerful tool for dividing polynomials by linear factors.
• Quadratic equations can be solved by factoring or using the quadratic formula.

Practice Makes Perfect

Finding the zeros of polynomial functions might seem tricky at first, but with practice, you'll become a pro! Try working through similar problems, and don't hesitate to ask for help if you get stuck. Remember, understanding these concepts is crucial for success in algebra and calculus. Keep up the great work, guys!