Polynomial Multiplication: Vertical Method Explained

Nov 15, 2025 by Alex Johnson 53 views

When you encounter problems like multiplying polynomials using the vertical method, it can seem a bit daunting at first, but it's actually a systematic approach that breaks down a complex task into manageable steps. Polynomial vertical multiplication is a powerful technique that helps organize the multiplication process, ensuring you don't miss any terms and correctly combine like terms. We'll walk through an example to make this clear.

Let's tackle the problem: Multiply $(x^2+5x+1)$ by $(3x^2-2x+4)$ using the vertical method. The key to this method is to align the polynomials vertically, much like you would with regular number multiplication. Think of it as multiplying each term in the top polynomial by each term in the bottom polynomial, but with a structured layout. We write the polynomial with more terms on top, although in this case, both have three terms. So, let's set it up:

  x^2 + 5x + 1
x 3x^2 - 2x + 4
-----------

The next step is to multiply the top polynomial by each term in the bottom polynomial, starting from the rightmost term of the bottom polynomial. So, we'll first multiply $(x^2+5x+1)$ by $+4$ .

Multiplying by the constant term (+4):

$4 * 1 = 4$
$4 * 5x = 20x$
$4 * x^2 = 4x^2$

This gives us the first line of our result: $4x^2 + 20x + 4$ . We write this below the line, aligning terms by their powers of x.

  x^2 + 5x + 1
x 3x^2 - 2x + 4
-----------
      4x^2 + 20x + 4

Next, we multiply the top polynomial $(x^2+5x+1)$ by the next term in the bottom polynomial, which is $-2x$ . Remember to pay close attention to the signs! When multiplying, we shift this result one place to the left, aligning terms by their powers of x. This is analogous to how you shift digits in standard multiplication.

Multiplying by the -2x term:

$-2x * 1 = -2x$
$-2x * 5x = -10x^2$
$-2x * x^2 = -2x^3$

So, we get $-2x^3 - 10x^2 - 2x$ . We write this below the first result, shifted one position to the left:

  x^2 + 5x + 1
x 3x^2 - 2x + 4
-----------
      4x^2 + 20x + 4 
    -2x^3 - 10x^2 -  2x

Finally, we multiply the top polynomial $(x^2+5x+1)$ by the leftmost term of the bottom polynomial, which is $3x^2$ . Again, we shift this result two places to the left.

Multiplying by the 3x^2 term:

$3x^2 * 1 = 3x^2$
$3x^2 * 5x = 15x^3$
$3x^2 * x^2 = 3x^4$

This gives us $3x^4 + 15x^3 + 3x^2$ . We add this to our calculation, shifted two positions to the left:

  x^2 + 5x + 1
x 3x^2 - 2x + 4
-----------
      4x^2 + 20x + 4 
    -2x^3 - 10x^2 -  2x
 3x^4 + 15x^3 +  3x^2

Now, the crucial step is to combine like terms by adding the columns vertically. Starting from the rightmost column (the constants):

Constant term: $4$
x term: $20x - 2x = 18x$
x^2 term: $4x^2 - 10x^2 + 3x^2 = (4 - 10 + 3)x^2 = -3x^2$
x^3 term: $-2x^3 + 15x^3 = (-2 + 15)x^3 = 13x^3$
x^4 term: $3x^4$

Putting it all together, we get the final result: $3x^4 + 13x^3 - 3x^2 + 18x + 4$ .

Let's re-evaluate the provided options based on our calculation. It seems there might have been a slight miscalculation in the manual process or the options provided. Let's double-check our multiplication and addition.

Let's re-do the calculation carefully:

$(x^2+5x+1) * (3x^2-2x+4)$

Multiply $(x^2+5x+1)$ by $4$ : $4x^2 + 20x + 4$
Multiply $(x^2+5x+1)$ by $-2x$ : $-2x^3 - 10x^2 - 2x$
Multiply $(x^2+5x+1)$ by $3x^2$ : $3x^4 + 15x^3 + 3x^2$

Now, sum these results vertically:

        3x^4 + 15x^3 +  3x^2
              -2x^3 - 10x^2 -  2x
                     4x^2 + 20x + 4
----------------------------------
  3x^4 + (15-2)x^3 + (3-10+4)x^2 + (20-2)x + 4
  3x^4 + 13x^3 + (-3)x^2 + 18x + 4

So the result is $3x^4 + 13x^3 - 3x^2 + 18x + 4$ .

Let's compare this to the given options:

A. $3 x^4+x^3+x^2+18 x+4$ B. $3 x^4-x^3+3 x^2+x+4$ C. $3 x^4-13 x^3+3 x^2+18 x+4$

It appears that none of the options perfectly match our calculated result. This can happen sometimes with textbook problems or online quizzes. Let's re-examine the multiplication process to ensure no errors.

Let's re-multiply the terms one more time:

Top polynomial: $P(x) = x^2+5x+1$ Bottom polynomial: $Q(x) = 3x^2-2x+4$

Multiply $P(x)$ by $4$ : $4(x^2+5x+1) = 4x^2+20x+4$

Multiply $P(x)$ by $-2x$ : $-2x(x^2+5x+1) = -2x^3 -10x^2 -2x$

Multiply $P(x)$ by $3x^2$ : $3x^2(x^2+5x+1) = 3x^4 + 15x^3 + 3x^2$

Adding these partial products:

        3x^4 + 15x^3 +  3x^2
              -2x^3 - 10x^2 -  2x
                     4x^2 + 20x + 4
----------------------------------
  3x^4 + (15-2)x^3 + (3-10+4)x^2 + (-2+20)x + 4
  3x^4 + 13x^3 + (-3)x^2 + 18x + 4

Our result is consistently $3x^4 + 13x^3 - 3x^2 + 18x + 4$ . Given the options, it's possible there's a typo in the question or the options. However, if we had to choose the closest one or if there was a common mistake to look for, we'd re-trace our steps.

Let's consider if there was a mistake in copying the problem. For instance, if the first polynomial was $x^2-5x+1$ or the second had different signs. But assuming the problem is as stated, our calculation should be correct.

Let's review the given options again:

A. $3 x^4+x^3+x^2+18 x+4$ B. $3 x^4-x^3+3 x^2+x+4$ C. $3 x^4-13 x^3+3 x^2+18 x+4$

Notice that option C has $3x^4$ , $18x$ , and $+4$ , which are consistent with our result. The discrepancy is in the $x^3$ and $x^2$ terms. Our result has $+13x^3$ and $-3x^2$ . Option C has $-13x^3$ and $+3x^2$ . This suggests a sign error could have occurred when setting up the options or in a common student error.

To be absolutely sure, let's try multiplying using the distributive property (FOIL extended) to cross-check.

$(x^2+5x+1)(3x^2-2x+4) = x^2(3x^2-2x+4) + 5x(3x^2-2x+4) + 1(3x^2-2x+4)$

$= (3x^4 - 2x^3 + 4x^2) + (15x^3 - 10x^2 + 20x) + (3x^2 - 2x + 4)$

Now, group and combine like terms:

$x^4$ terms: $3x^4$ $x^3$ terms: $-2x^3 + 15x^3 = 13x^3$ $x^2$ terms: $4x^2 - 10x^2 + 3x^2 = -3x^2$ x terms: $20x - 2x = 18x$ Constant terms: $4$

Result: $3x^4 + 13x^3 - 3x^2 + 18x + 4$ .

Our calculation is confirmed by the distributive property method. It strongly indicates that there might be an error in the provided options. In a test scenario, you would double-check your work and then perhaps select the option that has the most terms correct or is closest, or ask for clarification. For this specific problem, if we assume there's a typo in the options, our computed answer is definitive.

Let's assume, for the sake of selecting an answer, that option C had a typo and intended to be the correct answer, or that there was a specific way the problem was meant to be interpreted leading to one of the options. However, based on standard mathematical procedures, $3x^4 + 13x^3 - 3x^2 + 18x + 4$ is the correct result.

If we had to pick the closest answer, option C matches the highest power term ( $3x^4$ ), the $x$ term ( $+18x$ ), and the constant term ( $+4$ ). The $x^3$ and $x^2$ terms are where it differs. It has $-13x^3$ instead of $+13x^3$ , and $+3x^2$ instead of $-3x^2$ . This suggests a potential sign error in the formulation of option C.

In mathematics, precision is key. The vertical multiplication method is designed to help achieve that precision by organizing the steps. It's like building a house – each brick (term multiplication) needs to be placed correctly, and then they are all assembled (combined like terms) into a solid structure (the final polynomial). When results don't match options, it's a good opportunity to review your understanding of the method and to practice careful calculation. Often, a small arithmetic error or a misplaced sign can lead to a different final answer.

Let's review the problem again to ensure no misinterpretation:

\begin{array}{r} x^2+5 x+1 \\ x 3 x^2-2 x+4 \\ \hline \end{array} $ Which implies $(x^2+5x+1) imes (3x^2-2x+4)$. Our repeated calculations confirm that $3x^4 + 13x^3 - 3x^2 + 18x + 4$ is the correct answer. Therefore, assuming the question and options are presented as intended, none of the provided options are correct. However, if this were a multiple-choice question in an exam, and assuming a typo exists in the options, option C is the closest to the correct answer, differing only in the coefficients of the $x^3$ and $x^2$ terms, and potentially sign errors. To further solidify understanding of polynomial multiplication, exploring resources on **algebraic manipulation** can be very helpful. Understanding how to add, subtract, and multiply polynomials is fundamental to many areas of mathematics, including calculus and linear algebra. For more practice and detailed explanations, you can refer to resources like **Khan Academy** or **MathWorld**. * **Khan Academy:** Offers a vast library of free educational videos, exercises, and articles on algebra topics, including polynomial operations. You can find specific lessons on polynomial multiplication using various methods. * **MathWorld:** A comprehensive online encyclopedia of mathematics, providing detailed definitions, theorems, and examples related to algebra and polynomial theory. It's an excellent resource for in-depth understanding and advanced topics.