Solving Polynomials: Remainder Theorem & Factorization

Hey math enthusiasts! Let's dive into a cool polynomial problem. We're given a polynomial expression, $a + 7x + bx^3$, and some juicy information about it. When we divide this expression by $(x + 2)$, the remainder we get is $-12$. And get this, one of the factors of this polynomial is $(x - 2)$. Our mission, should we choose to accept it, is to figure out the values of the mysterious constants, $a$ and $b$. Sounds like fun, right?

This problem elegantly blends two important concepts in algebra: the Remainder Theorem and factorization. The Remainder Theorem is like a super shortcut. It tells us that if we divide a polynomial, $f(x)$, by $(x - c)$, the remainder is simply $f(c)$. So, if we know the remainder when dividing by $(x + 2)$, we can use that information to create an equation. Also, if $(x - 2)$ is a factor of the polynomial, it means that when we plug in $x = 2$, the polynomial magically becomes zero. This is another piece of the puzzle that we can use to solve for our unknowns. Let’s break it down step-by-step.

Applying the Remainder Theorem

Okay, let's get our hands dirty and put the Remainder Theorem to work. We know that when we divide our polynomial, $f(x) = a + 7x + bx^3$, by $(x + 2)$, the remainder is $-12$. The Remainder Theorem tells us that this means $f(-2) = -12$. This is because $(x + 2)$ can be rewritten as $(x - (-2))$, so $c = -2$ in our case. So, let’s substitute $x = -2$ into our polynomial and set the expression equal to $-12$.

So, plugging in $x = -2$ gives us: $a + 7(-2) + b(-2)^3 = -12$. Simplifying this equation, we get $a - 14 - 8b = -12$. We can rearrange this to look a bit cleaner: $a - 8b = 2$. We've just created our first equation! We'll call this Equation 1. This equation relates $a$ and $b$, but we can't solve it directly because we have two unknowns and only one equation. We'll need another equation to fully crack the code. That’s where the information about the factor $(x - 2)$ comes in handy. The key takeaway here is that the Remainder Theorem gives us a direct link between the divisor, the remainder, and the value of the polynomial at a specific point. This allows us to translate a division problem into a more manageable algebraic equation.

We're making good progress, guys! We've got our first equation. Now, let’s see what we can do with the information about the factor.

Utilizing the Factor Theorem

Alright, let's tap into the power of the Factor Theorem. We're told that $(x - 2)$ is a factor of our polynomial, $f(x) = a + 7x + bx^3$. This means that if we plug in $x = 2$ into the polynomial, the result should be zero. Why? Because a factor divides evenly into the polynomial, leaving no remainder. So, $f(2) = 0$. Let's plug in $x = 2$ into our polynomial:

So, we get: $a + 7(2) + b(2)^3 = 0$. Simplifying this, we get $a + 14 + 8b = 0$. Let’s rearrange this to get $a + 8b = -14$. This is our second equation! We’ll call this Equation 2. Now we have a system of two equations with two unknowns, which we can solve using various methods, like substitution or elimination. We can now solve for the values of 'a' and 'b'.

We’re using the Factor Theorem, which states that $(x - c)$ is a factor of a polynomial, $f(x)$, if and only if $f(c) = 0$. This theorem is just a special case of the Remainder Theorem, and it simplifies the process of finding the roots (or zeros) of a polynomial. The ability to recognize and apply these theorems is crucial for solving polynomial problems effectively. Now, let’s move on to solving these equations.

Solving the System of Equations

Alright, it's time to put on our detective hats and solve this system of equations! We have two equations:

Equation 1: $a - 8b = 2$ Equation 2: $a + 8b = -14$

We can use the elimination method here, which is super convenient in this case. Notice that the coefficients of $b$ have opposite signs. If we add the two equations together, the $b$ terms will cancel out, leaving us with an equation involving only $a$. So, let’s add Equation 1 and Equation 2:

$(a - 8b) + (a + 8b) = 2 + (-14)$

This simplifies to $2a = -12$. Now, we can solve for $a$ by dividing both sides by 2: $a = -6$. Great! We've found the value of $a$.

Now that we know $a$, we can substitute this value back into either Equation 1 or Equation 2 to solve for $b$. Let's use Equation 1: $a - 8b = 2$. Substituting $a = -6$, we get: $-6 - 8b = 2$. Adding 6 to both sides, we get $-8b = 8$. Finally, dividing both sides by $-8$, we get $b = -1$. Awesome! We've found the value of $b$ too!

So, the values are $a = -6$ and $b = -1$. The process of solving a system of equations, whether it's by elimination, substitution, or any other method, is fundamental in algebra. It helps us find the values of multiple unknown variables, and it's a skill you'll use over and over again in mathematics and other fields. The goal is to isolate one variable in terms of the other, and then substitute that value back into one of the original equations.

Verification and Conclusion

Okay, guys, we’ve found our answers: $a = -6$ and $b = -1$. But, as good mathematicians, we should always double-check our work. Let’s plug these values back into our original polynomial to make sure everything works as expected.

Our original polynomial was $f(x) = a + 7x + bx^3$. With our values, this becomes $f(x) = -6 + 7x - x^3$. Let's check if the remainder is indeed $-12$ when we divide by $(x + 2)$:

$f(-2) = -6 + 7(-2) - (-2)^3 = -6 - 14 - (-8) = -6 - 14 + 8 = -12$. Success! The remainder checks out.

Now, let's verify if $(x - 2)$ is a factor by plugging in $x = 2$:

$f(2) = -6 + 7(2) - (2)^3 = -6 + 14 - 8 = 0$. Boom! It works.

We have successfully found the values of $a$ and $b$. We started with a polynomial, used the Remainder Theorem and the Factor Theorem to create a system of equations, solved those equations, and then verified our solution. This problem is a great example of how different concepts in algebra work together to solve a seemingly complex problem. Keep practicing, and you'll become a polynomial pro in no time! Remember, understanding the underlying principles makes the problem-solving journey so much more enjoyable.

In conclusion, we successfully applied the Remainder Theorem, the Factor Theorem, and the system of equation methods to discover that when the expression $a + 7x + bx^3$ is divided by $(x + 2)$, the remainder is -12, and one of its factors is $(x - 2)$, then $a = -6$ and $b = -1$. These values ensure that all conditions in the problem statement are satisfied. This journey through polynomial algebra showcases the interconnectedness of mathematical concepts and provides a comprehensive understanding of the topic.