Polynomial Division & Trinomial Degrees: Math Problems Solved

Oct 14, 2025 by ADMIN 62 views

Hey everyone! Let's dive into some interesting math problems involving polynomial division and trinomial degrees. We'll break down each problem step-by-step so you can easily follow along. Get ready to sharpen those math skills!

1. Polynomial Division: Finding the Quotient

Our first problem involves finding the quotient of two polynomials. Specifically, given that x ≠ 1, we need to find the quotient of (x⁵ + x⁴ + x³ + x² + x) / (x³ + x² + x + 1). This looks a bit intimidating at first, but don't worry, we'll tackle it together. The key here is to manipulate the expression to make the division simpler.

First, let's factor out an x from the numerator:

x(x⁴ + x³ + x² + x + 1)

Now our expression looks like this:

[x(x⁴ + x³ + x² + x + 1)] / (x³ + x² + x + 1)

Unfortunately, direct division isn't immediately obvious, so let's try to see if we can rewrite the numerator in a way that makes the denominator a factor. We can rewrite the numerator polynomial by adding and subtracting terms strategically. Notice that if we had x - 1 as a factor, we could potentially simplify things using polynomial long division or synthetic division principles, but let's try grouping first to see if that gets us anywhere.

Let's try polynomial long division. Divide x⁵ + x⁴ + x³ + x² + x by x³ + x² + x + 1:

x² - x + 1
x³+x²+x+1 | x⁵ + x⁴ + x³ + x² + x + 0
          -(x⁵ + x⁴ + x³ + x²)
          ----------------------
                   0 + 0 + 0 + 0 + x + 0
                   -(-x³ - x² - x - 1)  <-- Oops, this doesn't work neatly.

That didn't go as planned! Let's try a different approach. Sometimes, recognizing patterns is crucial in math. We can rewrite the numerator by adding and subtracting '1' to create a geometric series kind of pattern. We can try to manipulate the numerator to resemble the denominator.

x(x⁴ + x³ + x² + x + 1) = x[(x⁴ + x³ + x² + x + 1)]

Another strategy is to perform polynomial long division directly. Dividing x⁵ + x⁴ + x³ + x² + x by x³ + x² + x + 1:

        x² - x + 1
x³+x²+x+1 | x⁵ + x⁴ + x³ + x² + x
        -(x⁵ + x⁴ + x³ + x²)
        ----------------------
                0 + 0 + 0 + 0 + x
                -(-x³ - x² - x - 1)
                ----------------------
                        x³ + x² + 2x + 1
                        -(x³ + x² + x + 1)
                        ----------------------
                                x + 0

Therefore: x⁵ + x⁴ + x³ + x² + x = (x³ + x² + x + 1)(x² - x + 1) + x.

So,

(x⁵ + x⁴ + x³ + x² + x) / (x³ + x² + x + 1) = (x² - x + 1) + [x / (x³ + x² + x + 1)]

It seems we hit a snag again. Factoring out x initially was a good move, but the subsequent division didn't simplify as much as we hoped. Let's go back to the original expression and rethink our strategy.

Let’s express it as:

[x⁵ + x⁴ + x³ + x² + x] / [x³ + x² + x + 1]

We can perform polynomial long division directly:

 x² - x + 1
x³+x²+x+1 | x⁵ + x⁴ + x³ + x² + x + 0
 -(x⁵ + x⁴ + x³ + x²)
 ----------------------
 0 + 0 + 0 + 0 + x + 0
 -(-x³ - x² - x - 1)
 ----------------------
 x³ + x² + 2x + 1
 -(x³ + x² + x + 1)
 ----------------------
 x

Thus, x⁵ + x⁴ + x³ + x² + x = (x³ + x² + x + 1)(x² - x + 1) + x

So, the original expression simplifies to:

(x⁵ + x⁴ + x³ + x² + x) / (x³ + x² + x + 1) = x² - x + 1 + x / (x³ + x² + x + 1)

Since we cannot simplify it further, x² - x + 1 + x / (x³ + x² + x + 1) is indeed our quotient with a remainder. But if the question is seeking only the polynomial part of the quotient, then it is x² - x + 1.

Final Answer: x² - x + 1 + x/(x³ + x² + x + 1), or x² - x + 1 if only the polynomial part is desired.

2. Trinomials and Degrees: Justification with Example

Now, let's move on to the second part of the problem. We need to justify the statement: "We can write a trinomial having degree 7" by providing an example. A trinomial, guys, is simply a polynomial with three terms. The degree of a polynomial is the highest power of the variable in the polynomial.

The statement is absolutely true! To create a trinomial with a degree of 7, we simply need to construct a polynomial with three terms, where the highest power of the variable is 7. One straightforward example would be:

5x⁷ + 3x² + 2

In this trinomial:

The first term is 5x⁷, which has a degree of 7.
The second term is 3x², which has a degree of 2.
The third term is 2, which is a constant term and has a degree of 0.

Since the highest degree among the three terms is 7, the degree of the entire trinomial is 7. This clearly demonstrates that we can indeed write a trinomial with a degree of 7. You could come up with countless other examples just by changing the coefficients or the lower-degree terms, but the key is to have one term with x⁷.

Here are a few more examples to solidify the concept:

x⁷ + 4x⁵ - 9x:
- Degree 7 (because of the x⁷ term).
- It has three terms, so it is a trinomial.
-2x⁷ + x³ + 10:
- Degree 7 (because of the -2x⁷ term).
- Again, it is a trinomial because it has three terms.

Remember, the coefficients (the numbers in front of the variables) don't affect the degree of the polynomial. Only the exponents matter when determining the degree. Also, the terms can be in any order; what's important is that the highest power of the variable is 7 and that there are three terms in total.

Explanation: A trinomial is a polynomial expression consisting of exactly three terms. The degree of a polynomial is the highest power of its variable. Therefore, to create a trinomial with a degree of 7, you must have one term where the variable is raised to the power of 7, and two other terms can have any lower powers. The coefficients (the numbers multiplying the variable terms) can be any real numbers, as long as they are not zero for the terms you want to include in the trinomial.

Example: Consider the trinomial 3x⁷ + 2x² + 5. Here:

3x⁷ is a term with degree 7.
2x² is a term with degree 2.
5 is a constant term (degree 0).

Since the highest degree among these terms is 7, the degree of the trinomial is 7. This validates the statement that we can indeed write a trinomial having a degree of 7.

Final Answer: Yes, the statement is true. Example: 5x⁷ + 3x² + 2 is a trinomial of degree 7.

Conclusion

So there you have it! We've tackled a polynomial division problem and justified the existence of a trinomial with a degree of 7. These problems highlight the importance of algebraic manipulation and understanding the definitions of polynomials and their degrees. Keep practicing, and you'll become a math whiz in no time! Keep an eye out for more math adventures!