Polynomial Multiplication Made Easy

Hey guys! Ever stared at a massive polynomial expression and felt a shiver down your spine? You know, the kind with multiple terms all jumbled up, just waiting to be multiplied? Well, buckle up, because today we're diving deep into the wild world of polynomial multiplication. Specifically, we're going to tackle a beast of an expression: $\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)$. This isn't just about getting the right answer; it's about understanding the process, mastering the techniques, and eventually, owning these problems like the math wizards you are. We'll break down this multi-part multiplication step-by-step, explaining each move so you can apply it to any polynomial challenge thrown your way. Get ready to boost your math game and impress yourselves with your newfound algebraic prowess!

Step 1: Conquer the First Two Polynomials

Alright, let's start taming this beast! Our mission is to multiply $\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)$. The smartest way to approach this, and this is a key strategy for any complex polynomial multiplication, is to break it down into smaller, manageable steps. We're going to first multiply the first two expressions: $\left(7 x^2\right)$ and $\left(2 x^3+5\right)$. This part is relatively simple because we're multiplying a monomial (a single term) by a binomial (two terms). We'll use the distributive property, which basically means each term in the second expression gets multiplied by the term in the first expression. So, we take $7x^2$ and multiply it by $2x^3$, and then we take $7x^2$ and multiply it by $5$.

Remember your exponent rules, guys! When you multiply terms with the same base (like $x^2$ and $x^3$), you add their exponents. So, $x^2 \times x^3 = x^{2+3} = x^5$. Therefore, $7x^2 \times 2x^3 = (7 \times 2) \times (x^2 \times x^3) = 14x^5$. Next, we multiply $7x^2$ by $5$. Since $5$ doesn't have an $x$ term, it's straightforward: $7x^2 \times 5 = (7 \times 5) \times x^2 = 35x^2$.

Putting it all together, the result of multiplying $\left(7 x^2\right)\left(2 x^3+5\right)$ is $14x^5 + 35x^2$. See? Not so scary when you take it one step at a time! This new expression, $14x^5 + 35x^2$, is what we'll call our intermediate result. It's crucial to get this part right because any errors here will cascade into the next step. Always double-check your arithmetic and your exponent addition. We've successfully simplified the first two parts of our problem, bringing us closer to the final answer. This foundational step ensures accuracy as we move forward to tackle the final binomial.

Step 2: Multiply the Intermediate Result by the Remaining Polynomial

Awesome work on the first step, you guys! Now we have our intermediate result, which is $14x^5 + 35x^2$. Our original problem has now been simplified to multiplying this intermediate result by the last part of the original expression: $\left(x^2-4 x-9\right)$. So, the problem now looks like: $\left(14x^5 + 35x^2\right)\left(x^2-4 x-9\right)$. This is a classic case of multiplying a binomial by a trinomial (an expression with three terms). Again, we'll rely on the distributive property, but this time it's a bit more extensive.

Think of it like this: each term in the first polynomial ($14x^5$ and $35x^2$) needs to be multiplied by each term in the second polynomial ($x^2$, $-4x$, and $-9$). It’s like a multiplication dance where every member of one group dances with every member of the other group. Let's break it down systematically.

First, we take the first term of our intermediate result, $14x^5$, and multiply it by each term in the trinomial:

• $14x^5 \times x^2 = 14x^{5+2} = 14x^7$
• $14x^5 \times (-4x) = (14 \times -4) \times (x^5 \times x^1) = -56x^{5+1} = -56x^6$
• $14x^5 \times (-9) = (14 \times -9) \times x^5 = -126x^5$

Next, we take the second term of our intermediate result, $35x^2$, and multiply it by each term in the trinomial:

• $35x^2 \times x^2 = 35x^{2+2} = 35x^4$
• $35x^2 \times (-4x) = (35 \times -4) \times (x^2 \times x^1) = -140x^{2+1} = -140x^3$
• $35x^2 \times (-9) = (35 \times -9) \times x^2 = -315x^2$

Now, we have a collection of terms from all these multiplications: $14x^7$, $-56x^6$, $-126x^5$, $35x^4$, $-140x^3$, and $-315x^2$. The next crucial step is to combine these terms. This is where we look for like terms – terms that have the same variable raised to the same power. In this case, we are fortunate because all the resulting terms have different powers of $x$ (from $x^7$ down to $x^2$), so there are no like terms to combine. This means our result is already in its simplest, expanded form. This might seem tedious, but each multiplication and addition of exponents is vital for accuracy. This methodical approach prevents errors and ensures we're on the right track.

Step 3: Combine Like Terms and Finalize the Answer

We're in the home stretch, guys! After meticulously multiplying each term from our intermediate result $\left(14x^5 + 35x^2\right)$ by each term in the trinomial $\left(x^2-4 x-9\right)$, we ended up with the following expanded terms: $14x^7$, $-56x^6$, $-126x^5$, $35x^4$, $-140x^3$, and $-315x^2$. The final, and arguably most satisfying, step in polynomial multiplication is to combine like terms. This is where we gather all terms that share the same variable and exponent and add or subtract their coefficients.

In this particular problem, as we noted in the previous step, each of the terms we generated has a unique power of $x$. We have an $x^7$ term, an $x^6$ term, an $x^5$ term, an $x^4$ term, an $x^3$ term, and an $x^2$ term. Since there are no two terms with the exact same variable and exponent combination, there are no like terms to combine. This means the expanded expression we derived is already in its simplest form.

To present the final answer clearly, it's customary to write the polynomial in descending order of powers, starting with the highest exponent and moving down to the lowest. This standard format makes polynomials easier to read and compare. Looking at our generated terms:

• The highest power is $x^7$, so $14x^7$ comes first.
• Next is $x^6$, so we add $-56x^6$.
• Then comes $x^5$, so we add $-126x^5$.
• Followed by $x^4$, so we add $35x^4$.
• Next is $x^3$, so we add $-140x^3$.
• Finally, the lowest power is $x^2$, so we add $-315x^2$.

Putting it all together in descending order, the final expanded form of $\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)$ is: $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$. This is our grand finale, the fully expanded and simplified product of the original expression. It's a testament to careful application of the distributive property and exponent rules. Always remember to arrange your final answer in descending order of powers for clarity and convention. Great job, everyone!

Identifying the Correct Option

Now that we've painstakingly worked through the entire multiplication process and arrived at our final answer, let's compare it with the given options. Our calculated result is $14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$. We need to find the option that exactly matches this expression. Let's examine each one:

• A. $14 x^5-x^4-46 x^3-58 x^2-20 x-45$: This expression has terms with lower powers of $x$ than we expect, and it's missing higher powers like $x^7$ and $x^6$. It's clearly not our answer. The powers and coefficients don't align with our step-by-step derivation.
• B. $14 x^6-56 x^5-91 x^4-140 x^3-315 x^2$: This option has some familiar coefficients, but the highest power is $x^6$, whereas our result starts with $x^7$. The powers of the terms are also shifted, indicating a potential miscalculation or a different starting point. This doesn't match our fully expanded form.
• C. $14 x^7-56 x^6-126 x^5+35 x^4-140 x^3-315 x^2$: Let's check this one carefully. The first term is $14x^7$, matching ours. The second is $-56x^6$, also matching. The third is $-126x^5$, a match. The fourth is $+35x^4$, matching. The fifth is $-140x^3$, another match. And the last term is $-315x^2$, which also matches perfectly. This option aligns precisely with our derived answer.
• D. $14 x^{12}-182 x^6+35 x^4-455 x^2$: This option seems to be multiplying the highest powers together ($x^2 \times x^3 \times x^2 = x^{2+3+2} = x^7$, not $x^{12}$), and the other coefficients and terms are significantly different. This looks like a result of a misunderstanding of the multiplication process or perhaps an entirely different operation.

Therefore, after careful comparison, Option C is the correct product of the given polynomial expression. This highlights the importance of not only performing the calculations accurately but also double-checking your work against the provided choices. Sometimes, seeing familiar numbers in incorrect arrangements can be a distractor, so a thorough review is essential. Well done for following along and reaching the correct conclusion!