Connect with knowledgeable individuals and find the best answers at IDNLearn.com. Discover comprehensive answers from knowledgeable members of our community, covering a wide range of topics to meet all your informational needs.
Sagot :
Let's analyze each given expression to determine whether it is a polynomial and, if it is, to categorize it by the number of terms it has.
### Expression a) [tex]\(-13 x^2 y + 52 y\)[/tex]
1. Is it a polynomial?
To determine if this is a polynomial, we need to check if each term consists of variables raised to non-negative integer exponents and whether it involves only addition, subtraction, or multiplication.
- The first term is [tex]\(-13 x^2 y\)[/tex], where [tex]\(x\)[/tex] is raised to the power of 2 and [tex]\(y\)[/tex] is raised to the power of 1.
- The second term is [tex]\(52 y\)[/tex], where [tex]\(y\)[/tex] is raised to the power of 1.
Both terms meet the criteria of a polynomial.
2. Number of terms?
This expression has two terms: [tex]\(-13 x^2 y\)[/tex] and [tex]\(52 y\)[/tex]. So, it is a binomial.
### Expression b) [tex]\(2 a^2 - 3 b^2 + 16\)[/tex]
1. Is it a polynomial?
- The first term is [tex]\(2 a^2\)[/tex], where [tex]\(a\)[/tex] is raised to the power of 2 (a non-negative integer).
- The second term is [tex]\(-3 b^2\)[/tex], where [tex]\(b\)[/tex] is raised to the power of 2.
- The third term is [tex]\(16\)[/tex], which is a constant term.
Since all terms meet the criteria of a polynomial, this expression is a polynomial.
2. Number of terms?
This expression has three terms: [tex]\(2 a^2\)[/tex], [tex]\(-3 b^2\)[/tex], and [tex]\(16\)[/tex]. So, it is a trinomial.
### Expression c) [tex]\(y + y^2 + 12\)[/tex]
1. Is it a polynomial?
- The first term is [tex]\(y\)[/tex], where [tex]\(y\)[/tex] is raised to the power of 1.
- The second term is [tex]\(y^2\)[/tex], where [tex]\(y\)[/tex] is raised to the power of 2.
- The third term is [tex]\(12\)[/tex], which is a constant term.
Since all terms meet the criteria of a polynomial, this expression is a polynomial.
2. Number of terms?
This expression has three terms: [tex]\(y\)[/tex], [tex]\(y^2\)[/tex], and [tex]\(12\)[/tex]. So, it is a trinomial.
### Expression d) [tex]\(-\frac{1}{2} x y z^3\)[/tex]
1. Is it a polynomial?
This expression involves a term [tex]\(-\frac{1}{2} x y z^3\)[/tex], which might initially appear to be a polynomial term because it consists of variables [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] raised to integer powers. However, the coefficient is a fraction [tex]\(-\frac{1}{2}\)[/tex].
In polynomials, the coefficients are allowed to be any real numbers, including fractions. Hence, this expression still meets the criteria of a polynomial.
2. Number of terms?
This expression has only one term: [tex]\(-\frac{1}{2} x y z^3\)[/tex]. So, it is a monomial.
### Summary:
- a) [tex]\(-13 x^2 y + 52 y\)[/tex] is a polynomial and is a binomial.
- b) [tex]\(2 a^2 - 3 b^2 + 16\)[/tex] is a polynomial and is a trinomial.
- c) [tex]\(y + y^2 + 12\)[/tex] is a polynomial and is a trinomial.
- d) [tex]\(-\frac{1}{2} x y z^3\)[/tex] is a polynomial and is a monomial.
### Expression a) [tex]\(-13 x^2 y + 52 y\)[/tex]
1. Is it a polynomial?
To determine if this is a polynomial, we need to check if each term consists of variables raised to non-negative integer exponents and whether it involves only addition, subtraction, or multiplication.
- The first term is [tex]\(-13 x^2 y\)[/tex], where [tex]\(x\)[/tex] is raised to the power of 2 and [tex]\(y\)[/tex] is raised to the power of 1.
- The second term is [tex]\(52 y\)[/tex], where [tex]\(y\)[/tex] is raised to the power of 1.
Both terms meet the criteria of a polynomial.
2. Number of terms?
This expression has two terms: [tex]\(-13 x^2 y\)[/tex] and [tex]\(52 y\)[/tex]. So, it is a binomial.
### Expression b) [tex]\(2 a^2 - 3 b^2 + 16\)[/tex]
1. Is it a polynomial?
- The first term is [tex]\(2 a^2\)[/tex], where [tex]\(a\)[/tex] is raised to the power of 2 (a non-negative integer).
- The second term is [tex]\(-3 b^2\)[/tex], where [tex]\(b\)[/tex] is raised to the power of 2.
- The third term is [tex]\(16\)[/tex], which is a constant term.
Since all terms meet the criteria of a polynomial, this expression is a polynomial.
2. Number of terms?
This expression has three terms: [tex]\(2 a^2\)[/tex], [tex]\(-3 b^2\)[/tex], and [tex]\(16\)[/tex]. So, it is a trinomial.
### Expression c) [tex]\(y + y^2 + 12\)[/tex]
1. Is it a polynomial?
- The first term is [tex]\(y\)[/tex], where [tex]\(y\)[/tex] is raised to the power of 1.
- The second term is [tex]\(y^2\)[/tex], where [tex]\(y\)[/tex] is raised to the power of 2.
- The third term is [tex]\(12\)[/tex], which is a constant term.
Since all terms meet the criteria of a polynomial, this expression is a polynomial.
2. Number of terms?
This expression has three terms: [tex]\(y\)[/tex], [tex]\(y^2\)[/tex], and [tex]\(12\)[/tex]. So, it is a trinomial.
### Expression d) [tex]\(-\frac{1}{2} x y z^3\)[/tex]
1. Is it a polynomial?
This expression involves a term [tex]\(-\frac{1}{2} x y z^3\)[/tex], which might initially appear to be a polynomial term because it consists of variables [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] raised to integer powers. However, the coefficient is a fraction [tex]\(-\frac{1}{2}\)[/tex].
In polynomials, the coefficients are allowed to be any real numbers, including fractions. Hence, this expression still meets the criteria of a polynomial.
2. Number of terms?
This expression has only one term: [tex]\(-\frac{1}{2} x y z^3\)[/tex]. So, it is a monomial.
### Summary:
- a) [tex]\(-13 x^2 y + 52 y\)[/tex] is a polynomial and is a binomial.
- b) [tex]\(2 a^2 - 3 b^2 + 16\)[/tex] is a polynomial and is a trinomial.
- c) [tex]\(y + y^2 + 12\)[/tex] is a polynomial and is a trinomial.
- d) [tex]\(-\frac{1}{2} x y z^3\)[/tex] is a polynomial and is a monomial.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
