Math Problem: Solving Cubic Equations

Jan 24, 2026 by Andrew McMorgan 38 views

Hey guys, today we're diving deep into the fascinating world of algebra, specifically tackling a cubic equation. You know, those polynomial beasts with the highest power being three? Yeah, those can be a bit tricky, but don't sweat it! We're going to break down this problem: solve for x in $3 x^3+x^2-10 x-8=0$ . We've already been given a head start: one of the roots is $x=- rac{4}{3}$ . This is super helpful because it means $(3x+4)$ is a factor of our polynomial. Our mission, should we choose to accept it, is to find the remaining roots. Let's get this party started!

Understanding Cubic Equations and Roots

Before we jump into solving, let's quickly recap what cubic equations and their roots are all about. A cubic equation is a polynomial equation of the form $ax^3 + bx^2 + cx + d = 0$ , where 'a' is not zero. The 'roots' of this equation are the values of 'x' that make the equation true, meaning when you plug them in, the whole thing equals zero. For a cubic equation, there will always be three roots, although they might not all be real numbers; some could be complex. Finding these roots is a classic math challenge, and there are several methods to approach it. Since we're given one root, $x = -4/3$ , we can use this information to simplify our problem. When we know a root, say 'r', then $(x-r)$ is a factor of the polynomial. In our case, since $x = -4/3$ is a root, $(x - (-4/3))$ is a factor. Multiplying this by 3 to get rid of the fraction, we get $(3x + 4)$ as a factor of $3x^3 + x^2 - 10x - 8$ . This is a crucial piece of information because it allows us to use polynomial division or synthetic division to reduce the cubic equation into a quadratic equation, which is much easier to solve. Remember, the goal is to find all the values of 'x' that satisfy the equation. Knowing one root is like finding a key that unlocks a big part of the puzzle. So, let's use this key to find the remaining pieces.

Using the Given Root to Factor the Polynomial

Alright, so we know that $(3x+4)$ is a factor of $3x^3 + x^2 - 10x - 8$ . The next logical step is to divide our cubic polynomial by this factor. This process will reveal the other factor, which will be a quadratic polynomial (degree 2). We can use either polynomial long division or synthetic division for this. Synthetic division is usually quicker if you're comfortable with it, especially when the divisor is linear. Let's go with synthetic division because it's pretty slick. We set up the synthetic division using the coefficients of the polynomial (3, 1, -10, -8) and the root of the divisor $(3x+4)$ , which is $x = -4/3$ .

Here's how synthetic division would look:

-4/3 | 3   1   -10   -8
    |     -4     4    8
    ------------------
      3  -3    -6    0

Let's break down what's happening here. We bring down the first coefficient (3). Then, we multiply it by the root $(-4/3)$ , which gives us -4. We add this to the next coefficient (1), resulting in -3. We multiply -3 by $-4/3$ to get 4, and add it to -10, giving us -6. Finally, we multiply -6 by $-4/3$ to get 8. Adding this to the last coefficient (-8) gives us 0. The fact that the remainder is 0 confirms that $(3x+4)$ is indeed a factor, and $x = -4/3$ is a root. The numbers in the bottom row (3, -3, -6) are the coefficients of the resulting quadratic polynomial. So, our original cubic equation can now be expressed as $(3x+4)(3x^2 - 3x - 6) = 0$ . Awesome, right? We've successfully reduced a cubic problem to a quadratic one!

Solving the Resulting Quadratic Equation

Now that we have factored our cubic equation into $(3x+4)(3x^2 - 3x - 6) = 0$ , we can find the remaining roots by solving the quadratic part: $3x^2 - 3x - 6 = 0$ . Before we dive into solving, we can simplify this quadratic equation by dividing all terms by 3. This makes the numbers smaller and easier to work with. So, we get $x^2 - x - 2 = 0$ . This is a much friendlier quadratic equation! There are a few ways to solve a quadratic equation: factoring, using the quadratic formula, or completing the square. Let's try factoring first, as it's often the quickest method if the quadratic is easily factorable. We are looking for two numbers that multiply to -2 and add up to -1 (the coefficient of the x term).

Think about it... the numbers are -2 and +1! Because $(-2) imes (+1) = -2$ and $(-2) + (+1) = -1$ . So, we can factor the quadratic as $(x-2)(x+1) = 0$ . To find the roots, we set each factor equal to zero:

$x - 2 = 0 ightarrow x = 2$
$x + 1 = 0 ightarrow x = -1$

And there you have it! We've found the other two roots of our original cubic equation. So, the roots of $3x^3 + x^2 - 10x - 8 = 0$ are $x = -4/3$ , $x = 2$ , and $x = -1$ . It's always a good idea to double-check these solutions by plugging them back into the original equation, but given our steps, we're pretty confident these are correct. This demonstrates how breaking down a complex problem into smaller, manageable parts, like using a known root to factor a polynomial, can lead to a straightforward solution.

Verifying the Roots

To be absolutely sure we've nailed this problem, let's take a moment to verify our roots. This is a crucial step in solving for x and ensuring accuracy in any mathematical problem. We've found three potential roots for the equation $3 x^3+x^2-10 x-8=0$ : $x=- rac{4}{3}$ , $x=2$ , and $x=-1$ . Let's plug each of these back into the original equation and see if they indeed result in zero.

First, let's check the given root, $x = - rac{4}{3}$ .

$3(- rac{4}{3})^3 + (- rac{4}{3})^2 - 10(- rac{4}{3}) - 8$ $= 3(- rac{64}{27}) + ( rac{16}{9}) + rac{40}{3} - 8$ $= - rac{64}{9} + rac{16}{9} + rac{120}{9} - rac{72}{9}$ $= rac{-64 + 16 + 120 - 72}{9}$ $= rac{-20}{9} eq 0$

Wait a minute! It seems there was a calculation error in the verification. Let's re-calculate that first step carefully.

$3(- rac{4}{3})^3 + (- rac{4}{3})^2 - 10(- rac{4}{3}) - 8$ $= 3(- rac{64}{27}) + ( rac{16}{9}) + rac{40}{3} - 8$ $= - rac{192}{27} + rac{16}{9} + rac{40}{3} - 8$ Ah, $3 imes (- rac{64}{27})$ is actually $- rac{64}{9}$ . Let's restart that calculation properly:

$3(- rac{4}{3})^3 + (- rac{4}{3})^2 - 10(- rac{4}{3}) - 8$ $= 3(- rac{64}{27}) + rac{16}{9} + rac{40}{3} - 8$ $= - rac{64}{9} + rac{16}{9} + rac{120}{9} - rac{72}{9}$ $= rac{-64 + 16 + 120 - 72}{9}$ $= rac{136 - 136}{9} = rac{0}{9} = 0$ .

Phew! Okay, the first root $x = -4/3$ is confirmed. My apologies for the slight hiccup there, guys – even mathematicians make calculation errors sometimes! It just goes to show the importance of careful checking.

Now, let's check $x = 2$ :

$3(2)^3 + (2)^2 - 10(2) - 8$ $= 3(8) + 4 - 20 - 8$ $= 24 + 4 - 20 - 8$ $= 28 - 28 = 0$ .

Excellent! $x=2$ is also a valid root.

Finally, let's check $x = -1$ :

$3(-1)^3 + (-1)^2 - 10(-1) - 8$ $= 3(-1) + 1 + 10 - 8$ $= -3 + 1 + 10 - 8$ $= -11 + 11 = 0$ .

Perfect! $x = -1$ is our third root.

All three roots, $x = -4/3$ , $x = 2$ , and $x = -1$ , satisfy the original equation. This rigorous verification process ensures that our factoring and solving steps were accurate. It's a fundamental part of mathematical problem-solving to confirm your answers, especially when dealing with potentially complex equations like this cubic one.

Conclusion: Mastering Cubic Equations

So, there you have it, math enthusiasts! We've successfully navigated the complexities of a cubic equation, $3 x^3+x^2-10 x-8=0$ , and found all its roots. The key to cracking this problem lay in leveraging the given root, $x=- rac{4}{3}$ , to factor the polynomial. By using synthetic division, we transformed the cubic equation into a more manageable quadratic equation, $x^2 - x - 2 = 0$ . This quadratic was then easily solved by factoring into $(x-2)(x+1)=0$ , yielding the additional roots $x=2$ and $x=-1$ . We then went the extra mile to verify each of these roots by plugging them back into the original equation, confirming that they all indeed make the equation equal to zero. This entire process, from initial setup to final verification, highlights the power of systematic approaches in algebra. Solving for x in cubic equations, or any polynomial for that matter, becomes much less daunting when you break it down into steps: identify knowns, apply relevant theorems or techniques (like factoring and polynomial division), solve the simplified forms, and finally, verify your solutions. Keep practicing these techniques, guys, and you'll be a cubic equation master in no time! Remember, every challenging math problem is just an opportunity to learn and grow your skills. Keep exploring, keep questioning, and keep solving!