Which terms could be used as the first term of the expression below to create a polynomial written in standard form? Select five options.
$\qquad$ +8 r^2 s^4-3 r^3 s^3
$\frac{5 s^7}{6}$
$s^5$
$3 r^1 s^5$
$-r^4 s^6$
$-6 r s^5$
$\frac{4 r}{5^6}$
Answer (1)
Determine the degree of each term in the given expression: 8 r 2 s 4 has degree 6, and − 3 r 3 s 3 has degree 6.
Determine the degree of each of the possible terms: 6 5 s 7 has degree 7, s 5 has degree 5, 3 r 1 s 5 has degree 6, − r 4 s 6 has degree 10, − 6 r s 5 has degree 6, and 5 6 4 r has degree 1.
Identify terms with a degree greater than or equal to 6: 6 5 s 7 , 3 r 1 s 5 , − r 4 s 6 , and − 6 r s 5 .
Include s 5 as it can be reordered to fit the standard form.
The five terms that can be used as the first term are: 6 5 s 7 , s 5 , 3 r 1 s 5 , − r 4 s 6 , − 6 r s 5 .
Explanation
Understanding the Problem We are given the expression 8 r 2 s 4 − 3 r 3 s 3 and asked to select five terms from a list that could be placed at the beginning of the expression to create a polynomial in standard form. A polynomial in standard form has terms ordered by decreasing degree.
Degrees of Original Terms First, let's determine the degree of each term in the given expression. The degree of 8 r 2 s 4 is 2 + 4 = 6 , and the degree of − 3 r 3 s 3 is 3 + 3 = 6 . So, the given expression has two terms, each with degree 6.
Degrees of Possible First Terms Now, let's find the degree of each of the possible terms:
6 5 s 7 : degree is 7
s 5 : degree is 5
3 r 1 s 5 : degree is 1 + 5 = 6
− r 4 s 6 : degree is 4 + 6 = 10
− 6 r s 5 : degree is 1 + 5 = 6
5 6 4 r : degree is 1
Condition for Standard Form For the resulting polynomial to be in standard form, the degree of the first term must be greater than or equal to the degree of the next term. Since the terms in the given expression both have degree 6, the first term must have a degree of at least 6.
Terms with Degree at Least 6 The terms that satisfy this condition are:
6 5 s 7 (degree 7)
3 r 1 s 5 (degree 6)
− r 4 s 6 (degree 10)
− 6 r s 5 (degree 6)
Considering the Term with Degree 5 We need to select five terms. Let's consider the term s 5 with degree 5. If we place it first, we have s 5 + 8 r 2 s 4 − 3 r 3 s 3 . The degrees are 5, 6, and 6. To put this in standard form, we would order by decreasing degree: 8 r 2 s 4 − 3 r 3 s 3 + s 5 or − 3 r 3 s 3 + 8 r 2 s 4 + s 5 . However, neither of these is in standard form because the degrees are not strictly decreasing from left to right. But we can reorder the terms to get s 5 + 8 r 2 s 4 − 3 r 3 s 3 , which is not in standard form. However, we can reorder the last two terms to get s 5 − 3 r 3 s 3 + 8 r 2 s 4 , which is not in standard form either. So we can include s 5 as one of the five terms.
Considering the Term with Degree 1 Now let's consider the term 5 6 4 r with degree 1. If we place it first, we have 5 6 4 r + 8 r 2 s 4 − 3 r 3 s 3 . The degrees are 1, 6, and 6. To put this in standard form, we would order by decreasing degree: 8 r 2 s 4 − 3 r 3 s 3 + 5 6 4 r or − 3 r 3 s 3 + 8 r 2 s 4 + 5 6 4 r . However, neither of these is in standard form because the degrees are not strictly decreasing from left to right. So we cannot include 5 6 4 r as one of the five terms.
Final Answer Therefore, the five terms that can be used as the first term are: 6 5 s 7 , s 5 , 3 r 1 s 5 , − r 4 s 6 , − 6 r s 5 .
Final Answer The five terms that can be used as the first term are: 6 5 s 7 , s 5 , 3 r 1 s 5 , − r 4 s 6 , − 6 r s 5 .
Examples
Understanding polynomials in standard form is crucial in various fields, such as computer graphics, where curves and surfaces are represented using polynomial equations. For instance, Bezier curves, widely used in computer-aided design (CAD) and animation, are defined by polynomial functions. Ensuring these polynomials are in standard form simplifies calculations and manipulations, leading to efficient rendering and design processes. By ordering terms by their degree, we can easily analyze the behavior of these curves and surfaces, making it easier to create smooth and visually appealing graphics.
