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Sagot :
To solve the given problem step by step:
### Question 1: The coefficient of the greatest common factor (GCF) of the polynomial expression
To find the coefficient of the GCF of the polynomial expression [tex]\(54 x^2 y^3 + 18 x y^5 - 36 y^4\)[/tex], we need to find the greatest common divisor (GCD) of the coefficients of the terms:
- The coefficients are:
- [tex]\(54\)[/tex]
- [tex]\(18\)[/tex]
- [tex]\(-36\)[/tex]
1. First, find the GCD of [tex]\(54\)[/tex] and [tex]\(18\)[/tex]:
- The divisors of [tex]\(54\)[/tex] are [tex]\(1, 2, 3, 6, 9, 18, 27, 54\)[/tex].
- The divisors of [tex]\(18\)[/tex] are [tex]\(1, 2, 3, 6, 9, 18\)[/tex].
- The common divisors are [tex]\(1, 2, 3, 6, 9, 18\)[/tex].
- The greatest common divisor is [tex]\(18\)[/tex].
2. Next, find the GCD of [tex]\(18\)[/tex] and [tex]\(-36\)[/tex]:
- The divisors of [tex]\(-36\)[/tex] are [tex]\(1, 2, 3, 4, 6, 9, 12, 18, 36\)[/tex] (and their negatives).
- The common divisors here again include [tex]\(1, 2, 3, 6, 9, 18\)[/tex].
- The greatest common divisor remains [tex]\(18\)[/tex].
Therefore, the coefficient of the greatest common factor of the polynomial expression is [tex]\(18\)[/tex].
### Question 2: The degree of the polynomial expression
To determine the degree of the polynomial expression [tex]\(54 x^2 y^3 + 18 x y^5 - 36 y^4\)[/tex], we need to consider the sum of the degrees of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] for each term and select the highest sum:
1. For the term [tex]\(54 x^2 y^3\)[/tex]:
- The degree of [tex]\(x^2\)[/tex] is [tex]\(2\)[/tex].
- The degree of [tex]\(y^3\)[/tex] is [tex]\(3\)[/tex].
- The total degree is [tex]\(2 + 3 = 5\)[/tex].
2. For the term [tex]\(18 x y^5\)[/tex]:
- The degree of [tex]\(x\)[/tex] is [tex]\(1\)[/tex].
- The degree of [tex]\(y^5\)[/tex] is [tex]\(5\)[/tex].
- The total degree is [tex]\(1 + 5 = 6\)[/tex].
3. For the term [tex]\(-36 y^4\)[/tex]:
- The degree of [tex]\(y^4\)[/tex] is [tex]\(4\)[/tex].
- There is no [tex]\(x\)[/tex] term, so its degree is [tex]\(0\)[/tex].
- The total degree is [tex]\(0 + 4 = 4\)[/tex].
The degree of the polynomial expression is the highest among these values, which is [tex]\(6\)[/tex].
### Final Answers:
1. The coefficient of the greatest common factor of the polynomial expression is [tex]\(\boxed{18}\)[/tex].
2. The degree of the polynomial expression is [tex]\(\boxed{6}\)[/tex].
### Question 1: The coefficient of the greatest common factor (GCF) of the polynomial expression
To find the coefficient of the GCF of the polynomial expression [tex]\(54 x^2 y^3 + 18 x y^5 - 36 y^4\)[/tex], we need to find the greatest common divisor (GCD) of the coefficients of the terms:
- The coefficients are:
- [tex]\(54\)[/tex]
- [tex]\(18\)[/tex]
- [tex]\(-36\)[/tex]
1. First, find the GCD of [tex]\(54\)[/tex] and [tex]\(18\)[/tex]:
- The divisors of [tex]\(54\)[/tex] are [tex]\(1, 2, 3, 6, 9, 18, 27, 54\)[/tex].
- The divisors of [tex]\(18\)[/tex] are [tex]\(1, 2, 3, 6, 9, 18\)[/tex].
- The common divisors are [tex]\(1, 2, 3, 6, 9, 18\)[/tex].
- The greatest common divisor is [tex]\(18\)[/tex].
2. Next, find the GCD of [tex]\(18\)[/tex] and [tex]\(-36\)[/tex]:
- The divisors of [tex]\(-36\)[/tex] are [tex]\(1, 2, 3, 4, 6, 9, 12, 18, 36\)[/tex] (and their negatives).
- The common divisors here again include [tex]\(1, 2, 3, 6, 9, 18\)[/tex].
- The greatest common divisor remains [tex]\(18\)[/tex].
Therefore, the coefficient of the greatest common factor of the polynomial expression is [tex]\(18\)[/tex].
### Question 2: The degree of the polynomial expression
To determine the degree of the polynomial expression [tex]\(54 x^2 y^3 + 18 x y^5 - 36 y^4\)[/tex], we need to consider the sum of the degrees of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] for each term and select the highest sum:
1. For the term [tex]\(54 x^2 y^3\)[/tex]:
- The degree of [tex]\(x^2\)[/tex] is [tex]\(2\)[/tex].
- The degree of [tex]\(y^3\)[/tex] is [tex]\(3\)[/tex].
- The total degree is [tex]\(2 + 3 = 5\)[/tex].
2. For the term [tex]\(18 x y^5\)[/tex]:
- The degree of [tex]\(x\)[/tex] is [tex]\(1\)[/tex].
- The degree of [tex]\(y^5\)[/tex] is [tex]\(5\)[/tex].
- The total degree is [tex]\(1 + 5 = 6\)[/tex].
3. For the term [tex]\(-36 y^4\)[/tex]:
- The degree of [tex]\(y^4\)[/tex] is [tex]\(4\)[/tex].
- There is no [tex]\(x\)[/tex] term, so its degree is [tex]\(0\)[/tex].
- The total degree is [tex]\(0 + 4 = 4\)[/tex].
The degree of the polynomial expression is the highest among these values, which is [tex]\(6\)[/tex].
### Final Answers:
1. The coefficient of the greatest common factor of the polynomial expression is [tex]\(\boxed{18}\)[/tex].
2. The degree of the polynomial expression is [tex]\(\boxed{6}\)[/tex].
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