Finding Roots Of A Cubic Function Using The Remainder Theorem

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Hey guys! Today, we're diving into a cool math problem where we need to find all the roots of a cubic function. A root of a function is simply a value of x that makes the function equal to zero. We'll be using the Remainder Theorem, which is a super handy tool for this kind of problem. So, let's jump right in!

Understanding the Problem

So, we are given the cubic function $f(x) = 4x^3 - 4x^2 - 16x + 16$, and we know that $(x - 2)$ is one of its factors. This is a crucial piece of information because it tells us that $x = 2$ is one root of the function. Our mission, should we choose to accept it (and we do!), is to find the other roots. This is where the Remainder Theorem comes into play, along with a bit of polynomial division and factoring. Trust me, it’s not as scary as it sounds!

The Remainder Theorem: Your New Best Friend

The Remainder Theorem is like a secret weapon in our math arsenal. It states that if you divide a polynomial f(x) by $(x - c)$, the remainder is f(c). In our case, since we know $(x - 2)$ is a factor, that means when we divide $f(x)$ by $(x - 2)$, the remainder should be zero. Why? Because if it’s a factor, it divides evenly! This also implies that f(2) = 0, confirming that x = 2 is indeed a root.

Polynomial Long Division: Time to Divide and Conquer

Now comes the fun part: polynomial long division. We're going to divide our cubic function $f(x) = 4x^3 - 4x^2 - 16x + 16$ by $(x - 2)$. If you're a bit rusty on polynomial long division, don't worry; we'll walk through it step by step.

  1. Set up the division: Write the dividend ($4x^3 - 4x^2 - 16x + 16$) inside the division symbol and the divisor ($x - 2$) outside.
  2. Divide the first terms: Divide the first term of the dividend ($4x^3$) by the first term of the divisor (x). This gives us $4x^2$. Write this above the division symbol.
  3. Multiply: Multiply the quotient term we just found ($4x^2$) by the entire divisor $(x - 2)$. This gives us $4x^3 - 8x^2$.
  4. Subtract: Subtract the result from the corresponding terms in the dividend. This gives us $(4x^3 - 4x^2) - (4x^3 - 8x^2) = 4x^2$.
  5. Bring down the next term: Bring down the next term from the dividend (-16x) to form the new expression $4x^2 - 16x$.
  6. Repeat: Repeat the process. Divide $4x^2$ by x, which gives us +4x. Write this next to $4x^2$ in the quotient. Multiply 4x by $(x - 2)$ to get $4x^2 - 8x$. Subtract this from $4x^2 - 16x$ to get $-8x$.
  7. Bring down the last term: Bring down the last term (+16) to form the new expression $-8x + 16$.
  8. Final Repeat: Divide -8x by x, which gives us -8. Write this next to +4x in the quotient. Multiply -8 by $(x - 2)$ to get $-8x + 16$. Subtract this from $-8x + 16$ to get 0. A remainder of 0 is exactly what we expected, which confirms $(x-2)$ is a factor.

So, after performing the polynomial long division, we find that:

4x3−4x2−16x+16x−2=4x2+4x−8\frac{4x^3 - 4x^2 - 16x + 16}{x - 2} = 4x^2 + 4x - 8

Factoring the Quadratic: Finding the Remaining Roots

Now we're left with a quadratic equation: $4x^2 + 4x - 8$. To find the remaining roots, we need to solve this equation for x. First, let's simplify it by dividing the entire equation by 4:

x2+x−2=0x^2 + x - 2 = 0

Now, we can factor this quadratic. We're looking for two numbers that multiply to -2 and add up to 1. Those numbers are 2 and -1. So, we can factor the quadratic as:

(x+2)(x−1)=0(x + 2)(x - 1) = 0

To find the roots, we set each factor equal to zero:

  • x+2=0⇒x=−2x + 2 = 0 \Rightarrow x = -2

  • x−1=0⇒x=1x - 1 = 0 \Rightarrow x = 1

Putting It All Together: The Roots Revealed

So, we found three roots for the function $f(x) = 4x^3 - 4x^2 - 16x + 16$:

  • From the given factor $(x - 2)$, we have $x = 2$.
  • From factoring the quadratic, we found $x = -2$ and $x = 1$.

Therefore, all the roots of the function are $x = -2, x = 1$, and $x = 2$.

Conclusion

Awesome work, team! We successfully found all the roots of the cubic function using the Remainder Theorem, polynomial long division, and factoring. Remember, the Remainder Theorem is a powerful tool that can help us find roots and factors of polynomials. Polynomial long division might seem a bit intimidating at first, but with practice, you'll become a pro. And factoring quadratics is a skill that will come in handy in many math problems. Keep practicing, and you'll be solving these problems like a math wizard in no time!

So, to recap, we took a cubic function, used the knowledge of one factor, and then systematically broke down the problem to find all the roots. This process not only helps in finding solutions but also deepens our understanding of polynomial functions. Whether you're tackling homework or preparing for a test, remember these steps, and you'll be well-equipped to handle similar problems. Keep up the great work, and I'll catch you in the next math adventure!