Polynomial Missing Term: Degree 5, Coefficient 16

Let's dive into the world of polynomials, guys! We've got a polynomial with a missing term, and our mission is to find it and then figure out what that means for the whole expression. Here's the polynomial we're working with:

$\square$ $+13 x^6-11 x^3-9 x^2+5 x-2$

We know the missing term has a degree of 5 and a coefficient of 16. This means the missing term is $16x^5$. So, let's plug that into our polynomial and see what we get.

Finding the Missing Term

The main goal here is to pinpoint that missing piece of our polynomial puzzle. We're told that this term has a degree of 5 and a coefficient of 16. Remember, the degree of a term is the exponent of the variable, and the coefficient is the number multiplying that variable. So, if we put that together, we get:

$16x^5$

That's it! That's our missing term. Now, let's add it to the original polynomial to get the complete polynomial:

$16x^5 + 13x^6 - 11x^3 - 9x^2 + 5x - 2$

So, the complete polynomial is:

$16x^5 + 13x^6 - 11x^3 - 9x^2 + 5x - 2$.

Polynomials are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The degree of a polynomial is the highest power of the variable in the polynomial. A coefficient is the number that multiplies a variable. A term in a polynomial is a single algebraic expression, which can be a constant, a variable, or a product of a constant and one or more variables. The standard form of a polynomial requires writing the terms in descending order of their degrees.

Understanding Standard Form

Now that we've found our missing term and have the complete polynomial, let's talk about standard form. Polynomials are usually written in standard form, which means arranging the terms in descending order of their degrees. In other words, we start with the term that has the highest exponent and work our way down to the constant term (the term without a variable).

Looking at our polynomial:

$16x^5 + 13x^6 - 11x^3 - 9x^2 + 5x - 2$

We can see that it's not in standard form because the term with the highest degree ($13x^6$) is not the first term. To put it in standard form, we need to rearrange the terms:

$13x^6 + 16x^5 - 11x^3 - 9x^2 + 5x - 2$

Ah, that's better! Now our polynomial is in standard form. The term with the highest degree comes first, and the exponents decrease as we move from left to right.

Standard form makes it easier to identify the degree of the polynomial and to compare different polynomials. To put a polynomial in standard form, identify the term with the highest degree. This term should be written first. Then, find the term with the next highest degree and write it next, and so on. Continue this process until all terms are written in descending order of their degrees. If there are any like terms (terms with the same variable and exponent), combine them before writing the polynomial in standard form. For example, if you have $3x^2 + 5x^2$, combine them to get $8x^2$ before placing it in the standard form of the polynomial.

Describing the Polynomial

So, which statement best describes our polynomial?

The question asks which statement best describes the polynomial:

$16x^5 + 13x^6 - 11x^3 - 9x^2 + 5x - 2$

Or, in standard form:

$13x^6 + 16x^5 - 11x^3 - 9x^2 + 5x - 2$

Let's analyze the options:

A. It is not in standard form because the degree of the first term is not the highest.

This statement is correct before we put the polynomial in standard form. However, after rearranging the terms, it is in standard form. So, this statement is not the best description of the polynomial after finding the missing term and rearranging.

To determine the best description of the polynomial, it is essential to consider its properties and characteristics. Let's examine a few key aspects:

Degree of the Polynomial: The degree of the polynomial is the highest power of the variable present in the expression. In our case, the highest power is 6 (from the term $13x^6$). Therefore, the degree of the polynomial is 6.

Leading Coefficient: The leading coefficient is the coefficient of the term with the highest degree. In our polynomial, the leading coefficient is 13 (from the term $13x^6$).

Number of Terms: The number of terms in the polynomial is the count of individual expressions separated by addition or subtraction. In our polynomial, we have 6 terms: $13x^6$, $16x^5$, $-11x^3$, $-9x^2$, $5x$, and $-2$.

Constant Term: The constant term is the term that does not contain any variables. In our polynomial, the constant term is -2.

Final Thoughts

So, there you have it! We found the missing term, put the polynomial in standard form, and described its key features. Polynomials might seem intimidating at first, but once you break them down, they're really just a collection of terms with different degrees and coefficients. Keep practicing, and you'll become a polynomial pro in no time!