Finding Zeros: Polynomial Function F(x) = 9x^3 + 9x^2 - 16x + 4

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Finding All Zeros of f(x) = 9x^3 + 9x^2 - 16x + 4

Hey guys! Let's dive into finding all the zeros of a polynomial function. Today, we're tackling the function f(x) = 9x^3 + 9x^2 - 16x + 4. The cool part is, we know it has at least one rational zero, which makes our job a whole lot easier. So, buckle up, and let’s get started!

Understanding Rational Zeros

First things first, what's a rational zero? Simply put, it's a zero of the polynomial that can be expressed as a fraction p/q, where p and q are integers. To find these rational zeros, we'll use the Rational Root Theorem. This theorem is a lifesaver, trust me! It narrows down the possibilities, so we're not just guessing blindly.

The Rational Root Theorem states that if a polynomial has integer coefficients, then every rational zero of the polynomial has the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In our case, the constant term is 4, and the leading coefficient is 9. So, the possible values for p are ±1, ±2, and ±4 (factors of 4), and the possible values for q are ±1, ±3, and ±9 (factors of 9).

This means our potential rational zeros are: ±1, ±2, ±4, ±1/3, ±2/3, ±4/3, ±1/9, ±2/9, and ±4/9. That’s quite a list, but don’t worry, we won’t have to test them all. We'll use a combination of synthetic division and a little bit of intuition to narrow it down.

Applying the Rational Root Theorem

Okay, let's get our hands dirty! We need to test these potential rational zeros. We can do this by plugging them into the function and seeing if the result is zero. Alternatively, we can use synthetic division, which is often a faster and more efficient method. Synthetic division not only tells us if a number is a zero but also gives us the quotient polynomial, which can help us find the remaining zeros.

Let's start with an easy one, like 1/3. We'll set up our synthetic division like this:

1/3 | 9  9  -16  4
      |________________

Bring down the 9, multiply by 1/3 to get 3, add it to 9 to get 12, multiply 12 by 1/3 to get 4, add it to -16 to get -12, multiply -12 by 1/3 to get -4, and add it to 4 to get 0. Bingo! A remainder of 0 means 1/3 is a zero of the function.

The synthetic division looks like this:

1/3 | 9  9  -16  4
      |    3  4  -4
      |________________
      9 12 -12  0

This also tells us that f(x) can be written as (x - 1/3)(9x^2 + 12x - 12). Now we have a quadratic equation to solve, which is much easier than a cubic!

Solving the Quadratic Equation

Now we need to find the zeros of the quadratic 9x^2 + 12x - 12. First, let's simplify it by dividing all the coefficients by their greatest common divisor, which is 3. This gives us 3x^2 + 4x - 4 = 0.

We can solve this quadratic equation using a few methods: factoring, completing the square, or the quadratic formula. Factoring is often the quickest if it's possible. Let's see if we can factor this one. We are looking for two numbers that multiply to (3)(-4) = -12 and add up to 4. Those numbers are 6 and -2.

So we can rewrite the middle term as 6x - 2x:

3x^2 + 6x - 2x - 4 = 0

Now, factor by grouping:

3x(x + 2) - 2(x + 2) = 0

(3x - 2)(x + 2) = 0

Setting each factor equal to zero gives us the remaining zeros:

3x - 2 = 0 => x = 2/3 x + 2 = 0 => x = -2

The Zeros of the Function

Alright, we've done it! We found all the zeros of the function f(x) = 9x^3 + 9x^2 - 16x + 4. They are 1/3, 2/3, and -2. How cool is that?

So, to recap, we used the Rational Root Theorem to narrow down the possible rational zeros, then used synthetic division to find one of the zeros, and finally factored the resulting quadratic equation to find the remaining zeros. It might seem like a lot of steps, but with practice, it becomes second nature.

Key Takeaways

  • The Rational Root Theorem is your best friend when finding rational zeros of polynomials.
  • Synthetic division is a powerful tool for testing potential zeros and simplifying the polynomial.
  • Don't forget to simplify quadratic equations before solving them – it can save you a lot of time and effort.
  • Always double-check your work, especially when dealing with signs and fractions.

I hope this explanation helped you understand how to find the zeros of a polynomial function. Keep practicing, and you'll become a pro in no time! If you have any questions, feel free to ask. Happy math-ing!

Practice Problems

To solidify your understanding, try these practice problems:

  1. Find all zeros of f(x) = 2x^3 - 5x^2 + 4x - 1.
  2. Find all zeros of f(x) = x^3 + 2x^2 - 5x - 6.

Good luck, and remember, math is awesome!

Conclusion

In conclusion, finding the zeros of a polynomial function like f(x) = 9x^3 + 9x^2 - 16x + 4 involves a systematic approach using the Rational Root Theorem and synthetic division. The Rational Root Theorem helps us identify potential rational zeros by considering the factors of the constant term and the leading coefficient. Synthetic division then allows us to efficiently test these potential zeros and reduce the polynomial to a lower degree, making it easier to solve. By breaking down the problem into manageable steps, we can confidently find all the zeros and gain a deeper understanding of polynomial functions. Remember, guys, practice makes perfect, so keep honing your skills!

This process not only helps in solving mathematical problems but also enhances analytical and problem-solving skills that are valuable in various fields. Whether you're a student tackling algebra problems or someone interested in the broader applications of mathematics, mastering these techniques will undoubtedly be beneficial. Keep exploring, keep learning, and most importantly, keep enjoying the journey of mathematical discovery! Understanding the zeros of polynomial functions is a fundamental concept in algebra, and it opens doors to more advanced topics in mathematics and beyond. So, let's embrace the challenge and continue to expand our knowledge and skills in this exciting realm.