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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.
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