Multiply Polynomials: (3x+5)(x^2-2x+4) Explained

When we talk about the product of two polynomials, we're essentially looking at what happens when you multiply them together. It might sound a bit daunting at first, especially with expressions like $(3x+5)(x^2-2x+4)$, but it's a fundamental skill in algebra. Think of it like distributing numbers in a multiplication problem, but instead of single numbers, we're dealing with terms that have variables and exponents. Our goal today is to break down how to find the product of $(3x+5)$ and $(x^2-2x+4)$ step-by-step, making sure you understand each part of the process. We'll cover the distributive property, combining like terms, and ensuring you can confidently tackle similar problems in the future. This skill is super useful, not just for math class, but it pops up in various fields like physics, engineering, and economics where understanding how different variables interact is key.

Understanding the Basics of Polynomial Multiplication

Before we dive into the specific problem of finding the product of two polynomials $(3x+5)$ and $(x^2-2x+4)$, let's quickly recap what polynomials are and the rules we follow when multiplying them. A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, $x^2-2x+4$ is a polynomial, and $3x+5$ is another. When we multiply polynomials, the key principle we use is the distributive property. This means that every term in the first polynomial must be multiplied by every term in the second polynomial. It sounds like a lot of steps, but breaking it down systematically makes it manageable. For instance, if you have $(a+b)(c+d)$, you multiply $a$ by $c$, $a$ by $d$, $b$ by $c$, and $b$ by $d$, resulting in $ac + ad + bc + bd$. This same concept applies, no matter how many terms are in each polynomial. Another crucial step after performing all the multiplications is to combine like terms. Like terms are terms that have the same variables raised to the same power. For example, $3x^2$ and $-5x^2$ are like terms because they both have the variable $x$ raised to the power of 2. We can add or subtract their coefficients to simplify the expression. So, the overall process involves distributing each term of the first polynomial to each term of the second, and then simplifying the result by combining any like terms. We'll apply these fundamental rules to our specific example to find the product of $(3x+5)(x^2-2x+4)$.

Step-by-Step Calculation of the Polynomial Product

Let's get down to business and calculate the product of the two polynomials: $(3x+5)(x^2-2x+4)$. We'll use the distributive property, which means we'll multiply each term in the first polynomial $(3x+5)$ by each term in the second polynomial $(x^2-2x+4)$. It's helpful to think of this as two separate distribution steps. First, we'll distribute the $3x$ from the first polynomial to all terms in the second polynomial. Then, we'll distribute the $5$ from the first polynomial to all terms in the second polynomial.

Step 1: Distribute the first term ($3x$)

• Multiply $3x$ by $x^2$: $3x imes x^2 = 3x^{(1+2)} = 3x^3$. Remember that when multiplying variables with exponents, you add the exponents.
• Multiply $3x$ by $-2x$: $3x imes (-2x) = -6x^{(1+1)} = -6x^2$. Be careful with the signs!
• Multiply $3x$ by $4$: $3x imes 4 = 12x$. This is a straightforward multiplication.

So, the first part of our distribution gives us: $3x^3 - 6x^2 + 12x$.

Step 2: Distribute the second term ($5$)

• Multiply $5$ by $x^2$: $5 imes x^2 = 5x^2$.
• Multiply $5$ by $-2x$: $5 imes (-2x) = -10x$. Again, mind the negative sign.
• Multiply $5$ by $4$: $5 imes 4 = 20$. This is a simple product.

So, the second part of our distribution gives us: $5x^2 - 10x + 20$.

Step 3: Combine the results from both distributions

Now, we need to put all the terms we got from both steps together:

$(3x^3 - 6x^2 + 12x) + (5x^2 - 10x + 20)$

To simplify this, we look for like terms and combine them. Like terms are those with the same variable raised to the same power.

• $x^3$ terms: We only have one $x^3$ term: $3x^3$.
• $x^2$ terms: We have $-6x^2$ and $5x^2$. Combine them: $-6x^2 + 5x^2 = -1x^2 = -x^2$.
• $x$ terms: We have $12x$ and $-10x$. Combine them: $12x - 10x = 2x$.
• Constant terms: We only have one constant term: $20$.

Step 4: Write the final product

Putting all the combined terms together in descending order of their exponents, we get our final answer for the product of the two polynomials:

$3x^3 - x^2 + 2x + 20$

This detailed breakdown shows how to systematically multiply polynomials using the distributive property and then simplify by combining like terms. This method is reliable for any polynomial multiplication problem you encounter.

Alternative Method: Vertical Multiplication

While the distributive method is excellent for understanding the core concept of polynomial multiplication, another way to find the product of two polynomials is through vertical multiplication. This method can be particularly helpful for larger or more complex polynomials, as it helps keep the terms organized and makes combining like terms more straightforward. It's similar to how you might multiply two multi-digit numbers by hand. For our example, $(3x+5)(x^2-2x+4)$, we would set up the multiplication vertically.

First, write the polynomials one above the other. It's usually best to put the polynomial with more terms on top, so we'll write $x^2-2x+4$ above $3x+5$. Make sure to align the terms by their powers of $x$. If any terms are missing (like an $x^3$ term in $3x+5$), you can leave a space or use a placeholder like $0x^3$. In this case, both polynomials are complete for their respective degrees.

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

Now, we multiply the top polynomial by each term of the bottom polynomial, starting with the bottom term that has the lowest power (the constant term, which is 5 in this case). We multiply 5 by each term in the top polynomial and write the result on the first line.

• $5 imes 4 = 20$
• $5 imes (-2x) = -10x$
• $5 imes x^2 = 5x^2$

So, the first line of our product is: $5x^2 - 10x + 20$. (It's good practice to write these in descending order of powers of x, so $5x^2 - 10x + 20$).

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

Next, we multiply the top polynomial by the next term in the bottom polynomial, which is $3x$. When we write this result, we need to shift it one place to the left so that like terms are aligned vertically. This is the crucial step for organization in vertical multiplication.

• $3x imes 4 = 12x$
• $3x imes (-2x) = -6x^2$
• $3x imes x^2 = 3x^3$

So, the second line of our product is: $3x^3 - 6x^2 + 12x$. We write this under the first line, aligning the $x$ term with the $x$ term, the $x^2$ term with the $x^2$ term, and so on.

    x^2 - 2x + 4
  x        3x + 5
  -----------------
      5x^2 - 10x + 20 
  3x^3 - 6x^2 + 12x      

Now, we add the two lines together, column by column (combining like terms). Make sure to bring down any terms that don't have a corresponding term in the other line.

• The $x^3$ column has only $3x^3$, so we bring that down: $3x^3$.
• The $x^2$ column has $5x^2$ and $-6x^2$. Adding them gives: $5x^2 - 6x^2 = -x^2$. So, $-x^2$.
• The $x$ column has $-10x$ and $12x$. Adding them gives: $-10x + 12x = 2x$. So, $+2x$.
• The constant column has only $20$, so we bring that down: $+20$.

Putting it all together, we get the final product of the two polynomials:

$3x^3 - x^2 + 2x + 20$

As you can see, both the distributive method and the vertical multiplication method yield the same result. The vertical method often helps prevent errors by keeping terms neatly aligned. It's a great tool to have in your algebraic toolkit!

Why Polynomial Products Matter

Understanding how to calculate the product of two polynomials is a cornerstone of algebra, and its significance extends far beyond textbook exercises. When you master multiplying polynomials, you're not just learning a computational skill; you're grasping a fundamental concept that underpins many advanced mathematical and scientific applications. For instance, in calculus, when you need to find the derivative or integral of a complex function that is a product of polynomials, knowing how to expand that product simplifies the entire process. This leads to more accurate and efficient calculations. Think about physics, where polynomial functions are used to model trajectories, wave behaviors, and electrical circuits. If you're analyzing the interaction of two such phenomena, you might need to multiply their polynomial representations to understand their combined effect. This is where polynomial multiplication becomes directly applicable. In economics, polynomial functions can model cost, revenue, and profit. When businesses project how changes in production levels (represented by a polynomial) might affect overall profits (potentially another polynomial), multiplying these functions can reveal crucial insights into market dynamics and strategic decision-making. Furthermore, in computer graphics and engineering, polynomial curves (like Bezier curves) are used extensively for design. Understanding the mathematical operations, including multiplication, on these curves is essential for manipulating shapes and creating complex visual effects. The ability to find the product of polynomials is also a stepping stone to understanding more advanced algebraic structures and concepts, such as factoring and solving polynomial equations. These skills are vital for problem-solving in a wide array of fields. Therefore, the seemingly simple act of multiplying two polynomials is a powerful skill that unlocks deeper understanding and enables complex problem-solving across science, technology, engineering, and mathematics (STEM) fields and beyond. It's a building block that supports much of higher mathematics and its practical applications.

Conclusion: Mastering Polynomial Products

In conclusion, calculating the product of two polynomials, such as $(3x+5)(x^2-2x+4)$, is a fundamental algebraic skill that involves systematic application of the distributive property. We've explored two effective methods: the step-by-step distributive approach and the organized vertical multiplication. Both methods require careful attention to detail, especially when dealing with signs and combining like terms. The distributive method involves multiplying each term of the first polynomial by each term of the second and then summing the results, followed by simplification. The vertical method offers a structured way to perform the same multiplication, aligning like terms for easier addition. The resulting product, $3x^3 - x^2 + 2x + 20$, demonstrates the outcome of these operations. Remember that practice is key to mastering these techniques. The ability to confidently multiply polynomials opens doors to solving more complex algebraic problems and is crucial for numerous applications in mathematics, science, and engineering. Keep practicing, and you'll find these calculations become second nature!

For further exploration into polynomial operations and algebraic concepts, you can refer to resources like Khan Academy for detailed explanations and practice problems, or consult Paul's Online Math Notes for comprehensive mathematical resources.