Polynomial Operations F(x) And G(x) Explained

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Hey guys! 👋 Ever found yourself staring at polynomial equations and feeling like you're deciphering an alien language? Don't worry, you're not alone! Polynomials might seem intimidating at first, but once you grasp the basics, they become surprisingly manageable. Today, we're diving deep into the world of polynomial operations, taking on a specific problem to illustrate the key concepts. We'll break down each step in detail, making sure you not only get the answer but also understand the why behind it. So, grab your thinking caps, and let's get started!

Delving into Polynomials

Before we jump into the problem, let's take a moment to understand what polynomials actually are. In essence, polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Think of them as mathematical building blocks – they can be used to model a wide range of real-world phenomena, from the trajectory of a ball to the growth of a population.

The general form of a polynomial is: a_n*x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where:

  • x is the variable
  • a_n, a_{n-1}, ..., a_1, a_0 are the coefficients (constants)
  • n is a non-negative integer representing the degree of the term. The degree of the polynomial is the highest power of the variable in the expression.

For example, the expression 4x³ - 3x² + 7x - 6 is a polynomial. Here:

  • The coefficients are 4, -3, 7, and -6.
  • The variable is x.
  • The degrees of the terms are 3, 2, 1, and 0 (remember that a constant term has a degree of 0 since it can be written as x⁰).
  • The degree of the polynomial is 3 (the highest power of x).

Now that we have a solid understanding of what polynomials are, let's tackle the main problem!

The Challenge: Polynomial Operations with F(x) and G(x)

We're given two polynomials:

  • F(x) = 4x³ - 3x² + 7x - 6
  • G(x) = 4x³ + 2x² - 3x + 5

Our mission, should we choose to accept it (and we do!), is to determine the following:

a) F(x) + G(x) b) F(x) - G(x) c) F(x) * G(x) and its degree

Let's break down each part step-by-step.

a) Adding F(x) and G(x): Combining Like Terms

The first task is to find F(x) + G(x). This involves adding the two polynomials together. The key here is to combine like terms. Like terms are those that have the same variable raised to the same power. For instance, 4x³ and 4x³ are like terms, while 4x³ and -3x² are not.

Here's how we add F(x) and G(x):

F(x) + G(x) = (4x³ - 3x² + 7x - 6) + (4x³ + 2x² - 3x + 5)

Now, let's group the like terms together:

= (4x³ + 4x³) + (-3x² + 2x²) + (7x - 3x) + (-6 + 5)

Finally, we perform the addition:

= 8x³ - x² + 4x - 1

So, F(x) + G(x) = 8x³ - x² + 4x - 1. Easy peasy, right? We simply identified the like terms and added their coefficients. The degree of this resulting polynomial is 3, as that's the highest power of x.

b) Subtracting G(x) from F(x): Watch Out for the Signs!

Next, we need to find F(x) - G(x). Subtraction is similar to addition, but we need to be extra careful with the signs. Remember that subtracting a polynomial is the same as adding the negative of that polynomial. This means we need to distribute the negative sign to each term within G(x).

Let's set it up:

F(x) - G(x) = (4x³ - 3x² + 7x - 6) - (4x³ + 2x² - 3x + 5)

Distribute the negative sign:

= (4x³ - 3x² + 7x - 6) + (-4x³ - 2x² + 3x - 5)

Now, group the like terms:

= (4x³ - 4x³) + (-3x² - 2x²) + (7x + 3x) + (-6 - 5)

Perform the operations:

= 0x³ - 5x² + 10x - 11

Which simplifies to:

= -5x² + 10x - 11

Therefore, F(x) - G(x) = -5x² + 10x - 11. Notice how the degree of this polynomial is 2, as that's the highest power of x. Sign handling is crucial here; a simple mistake can change the entire result. So, always double-check your work!

c) Multiplying F(x) and G(x): The Distributive Dance

Now comes the fun part – multiplying F(x) and G(x). This is a bit more involved than addition or subtraction, but we can conquer it by using the distributive property. Basically, we need to multiply each term in F(x) by each term in G(x) and then combine like terms.

F(x) * G(x) = (4x³ - 3x² + 7x - 6) * (4x³ + 2x² - 3x + 5)

This looks daunting, but let's break it down. We'll systematically multiply each term in the first polynomial by each term in the second:

  • 4x³ * (4x³ + 2x² - 3x + 5) = 16x⁶ + 8x⁵ - 12x⁴ + 20x³
  • -3x² * (4x³ + 2x² - 3x + 5) = -12x⁵ - 6x⁴ + 9x³ - 15x²
  • 7x * (4x³ + 2x² - 3x + 5) = 28x⁴ + 14x³ - 21x² + 35x
  • -6 * (4x³ + 2x² - 3x + 5) = -24x³ - 12x² + 18x - 30

Now, we add all these results together:

16x⁶ + 8x⁵ - 12x⁴ + 20x³ - 12x⁵ - 6x⁴ + 9x³ - 15x² + 28x⁴ + 14x³ - 21x² + 35x - 24x³ - 12x² + 18x - 30

Time to combine those like terms! This is where things can get a bit messy, so take your time and be careful:

= 16x⁶ + (8x⁵ - 12x⁵) + (-12x⁴ - 6x⁴ + 28x⁴) + (20x³ + 9x³ + 14x³ - 24x³) + (-15x² - 21x² - 12x²) + (35x + 18x) - 30

Simplify:

= 16x⁶ - 4x⁵ + 10x⁴ + 19x³ - 48x² + 53x - 30

Phew! That was a workout. So, F(x) * G(x) = 16x⁶ - 4x⁵ + 10x⁴ + 19x³ - 48x² + 53x - 30.

And what's the degree of this polynomial? It's 6, the highest power of x.

Conclusion: Mastering Polynomial Operations

And there you have it! We've successfully tackled the problem, performing addition, subtraction, and multiplication of polynomials. We found that:

  • F(x) + G(x) = 8x³ - x² + 4x - 1
  • F(x) - G(x) = -5x² + 10x - 11
  • F(x) * G(x) = 16x⁶ - 4x⁵ + 10x⁴ + 19x³ - 48x² + 53x - 30

We also determined the degrees of the resulting polynomials in each operation.

The key takeaways from this exercise are:

  • Like Terms are Your Friends: When adding or subtracting polynomials, always combine like terms.
  • Signs Matter: Be extra careful with signs, especially during subtraction.
  • Distribute Diligently: For multiplication, systematically distribute each term in one polynomial to every term in the other.
  • Take Your Time: Polynomial operations can be lengthy, so don't rush. Double-check your work to avoid errors.

By mastering these fundamental operations, you'll be well-equipped to tackle more advanced polynomial problems. Keep practicing, and polynomials will become your mathematical allies! You got this!

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