Polynomial Divisibility Is 3x^5 + 7x^4 + 8x^3 - 2x^2 + 14x + 8 Divisible

Hey guys! Today, we're diving into the exciting world of polynomial divisibility. We've got a big polynomial here: $3x^5 + 7x^4 + 8x^3 - 2x^2 + 14x + 8$, and our mission is to figure out which of the following expressions divide it evenly. It's like a mathematical detective game, and we're the sleuths!

Our Suspects:

a. $2x - 2$ b. $2 - x$ c. $x + 3$ d. $3x + 4$

So, how do we crack this case? We're going to use a powerful tool called the Factor Theorem and some good old-fashioned polynomial division. Buckle up; it's going to be a fun ride!

The Factor Theorem: Our Secret Weapon

The Factor Theorem is a cornerstone concept in algebra, a true game-changer when it comes to polynomial factorization. Think of it as a mathematical decoder ring, allowing us to unveil the secrets hidden within polynomials. Essentially, the theorem forges a powerful connection between the roots of a polynomial and its factors. In simpler terms, it states: If a polynomial f(x) equals zero when x is set to a certain value a, then (x - a) is a factor of f(x). Conversely, if (x - a) is a factor of f(x), then f(a) must equal zero.

To truly grasp the essence of the Factor Theorem, let’s break down its significance and application through an example. Consider a polynomial, say, f(x) = x² - 5x + 6. Our mission is to determine if (x - 2) is a factor of this polynomial. According to the Factor Theorem, we simply need to evaluate f(2). Substituting x with 2, we get f(2) = (2)² - 5(2) + 6 = 4 - 10 + 6 = 0. The result is zero, which, according to the Factor Theorem, definitively tells us that (x - 2) is indeed a factor of f(x). This is more than just a mathematical trick; it's a fundamental insight that simplifies the process of polynomial factorization. Imagine trying to factor a high-degree polynomial without this tool – it would be like navigating a maze blindfolded. The Factor Theorem provides us with a clear path, allowing us to systematically identify factors and simplify complex expressions.

The beauty of the Factor Theorem lies not only in its simplicity but also in its versatility. It’s applicable to polynomials of any degree, making it an invaluable asset in algebraic manipulations. Whether you're solving equations, simplifying expressions, or tackling advanced mathematical problems, the Factor Theorem is a reliable companion. So, let’s remember this powerful tool as we continue our exploration into the realm of polynomials. It’s not just a theorem; it’s a key to unlocking the hidden structures within algebraic expressions, allowing us to solve problems with elegance and precision.

Cracking the Case: Applying the Factor Theorem

Now, let's put the Factor Theorem to work! Our polynomial is $f(x) = 3x^5 + 7x^4 + 8x^3 - 2x^2 + 14x + 8$. We'll go through each suspect, setting the potential factor equal to zero and solving for x. This will give us the value we need to plug into f(x).

Suspect A: $2x - 2$

• Set $2x - 2 = 0$
• Solve for x: $2x = 2$, so $x = 1$

Now, we evaluate $f(1)$:

$f(1) = 3(1)^5 + 7(1)^4 + 8(1)^3 - 2(1)^2 + 14(1) + 8 = 3 + 7 + 8 - 2 + 14 + 8 = 38$

Since $f(1) = 38$ and not 0, $2x - 2$ is not a factor.

Suspect B: $2 - x$

• Set $2 - x = 0$
• Solve for x: $x = 2$

Now, we evaluate $f(2)$:

$f(2) = 3(2)^5 + 7(2)^4 + 8(2)^3 - 2(2)^2 + 14(2) + 8 = 3(32) + 7(16) + 8(8) - 2(4) + 14(2) + 8 = 96 + 112 + 64 - 8 + 28 + 8 = 300$

Since $f(2) = 300$ and not 0, $2 - x$ is not a factor.

Suspect C: $x + 3$

• Set $x + 3 = 0$
• Solve for x: $x = -3$

Now, we evaluate $f(-3)$:

$f(-3) = 3(-3)^5 + 7(-3)^4 + 8(-3)^3 - 2(-3)^2 + 14(-3) + 8 = 3(-243) + 7(81) + 8(-27) - 2(9) + 14(-3) + 8 = -729 + 567 - 216 - 18 - 42 + 8 = -430$

Since $f(-3) = -430$ and not 0, $x + 3$ is not a factor.

Suspect D: $3x + 4$

• Set $3x + 4 = 0$
• Solve for x: $3x = -4$, so x = -rac{4}{3}

Now, we evaluate f(-rac{4}{3}). This one's going to be a bit trickier with the fraction, so let's be careful:

f(-rac{4}{3}) = 3(-rac{4}{3})^5 + 7(-rac{4}{3})^4 + 8(-rac{4}{3})^3 - 2(-rac{4}{3})^2 + 14(-rac{4}{3}) + 8

f(-rac{4}{3}) = 3(-rac{1024}{243}) + 7(rac{256}{81}) + 8(-rac{64}{27}) - 2(rac{16}{9}) + 14(-rac{4}{3}) + 8

f(-rac{4}{3}) = -rac{1024}{81} + rac{1792}{81} - rac{512}{27} - rac{32}{9} - rac{56}{3} + 8

To make things easier, let's get a common denominator of 81:

f(-rac{4}{3}) = -rac{1024}{81} + rac{1792}{81} - rac{1536}{81} - rac{288}{81} - rac{1512}{81} + rac{648}{81}

f(-rac{4}{3}) = rac{-1024 + 1792 - 1536 - 288 - 1512 + 648}{81} = rac{-1920}{81}

It seems we made a mistake in our calculations. Let's try polynomial long division instead for this suspect.

Polynomial Long Division: A Direct Approach

Sometimes, the Factor Theorem can be a bit cumbersome, especially with fractions. That's where polynomial long division comes in handy. It's a direct way to see if one polynomial divides another evenly.

Let's divide $3x^5 + 7x^4 + 8x^3 - 2x^2 + 14x + 8$ by $3x + 4$.

 x^4 + x^3 + rac{4}{3}x^2 - 2x + 6
3x + 4 | 3x^5 + 7x^4 + 8x^3 - 2x^2 + 14x + 8
        -(3x^5 + 4x^4)
        ------------------
               3x^4 + 8x^3
               -(3x^4 + 4x^3)
               ------------------
                      4x^3 - 2x^2
                      -(4x^3 + rac{4}{3}x^2)
                      ------------------
                             -rac{10}{3}x^2 + 14x
                             -(-rac{10}{3}x^2 - rac{40}{9}x)
                             ------------------
                                    rac{166}{9}x + 8
                                    -(rac{166}{9}x + rac{664}{27})
                                    ------------------
                                           8 - rac{664}{27} = -rac{448}{27}

Since we have a remainder (-rac{448}{27}), $3x + 4$ is not a factor.

The Verdict: Unmasking the Factors

After our investigation, using both the Factor Theorem and polynomial long division, we've come to a conclusion:

• $2x - 2$: Not a factor
• $2 - x$: Not a factor
• $x + 3$: Not a factor
• $3x + 4$: Not a factor

It turns out that none of the given expressions are factors of $3x^5 + 7x^4 + 8x^3 - 2x^2 + 14x + 8$. Sometimes, the mystery leads to an unexpected answer! Keep practicing, guys, and you'll become polynomial pros in no time!