If the degree of polynomial 9 x 5y 2z ris 15, then r = ?
8
Understanding Polynomial Degree
The question asks us to find the value of 'r' in the polynomial term \(9x^5y^2z^r\), given that the degree of this term is 15. To solve this, we need to understand what the degree of a polynomial term is.
A polynomial term (or monomial) consists of a coefficient (the number in front, like 9) and one or more variables raised to non-negative integer exponents. The degree of a single term is the sum of the exponents of all the variables in that term.
Calculating the Degree of the Term
The given polynomial term is \(9x^5y^2z^r\). The variables in this term are x, y, and z. Their respective exponents are:
- Exponent of x is 5
- Exponent of y is 2
- Exponent of z is r
The degree of the term is the sum of these exponents:
Degree = Exponent of x + Exponent of y + Exponent of z
Degree = \(5 + 2 + r\)
Setting up the Equation
We are given that the degree of the polynomial term \(9x^5y^2z^r\) is 15.
So, we can set up the equation:
\(5 + 2 + r = 15\)
Simplify the left side of the equation:
\(7 + r = 15\)
Solving for r
To find the value of r, we need to isolate r in the equation \(7 + r = 15\). We can do this by subtracting 7 from both sides of the equation:
\(r = 15 - 7\)
\(r = 8\)
So, the value of r is 8.
Verifying the Solution
If r = 8, the term is \(9x^5y^2z^8\). The degree of this term would be the sum of the exponents: \(5 + 2 + 8 = 7 + 8 = 15\). This matches the given degree, so our solution is correct.
Conclusion
The value of r that makes the degree of the polynomial term \(9x^5y^2z^r\) equal to 15 is 8.
Revision Table: Key Polynomial Concepts
| Concept | Definition | Example |
|---|---|---|
| Polynomial Term (Monomial) | A single term consisting of a coefficient and variables raised to non-negative integer exponents. | \(4x^3\), \(-7y^5z\), \(9\) |
| Degree of a Monomial | The sum of the exponents of all variables in the term. | Degree of \(4x^3\) is 3. Degree of \(-7y^5z^1\) is \(5+1=6\). Degree of \(9\) (or \(9x^0\)) is 0. |
| Polynomial | An expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. | \(3x^2 + 2x - 5\), \(y^4 - 7y + 1\) |
| Degree of a Polynomial | The highest degree among all the terms in the polynomial. | Degree of \(3x^2 + 2x - 5\) is 2 (highest degree among terms \(3x^2\) (degree 2), \(2x\) (degree 1), \(-5\) (degree 0)). |
Additional Information on Polynomials
Polynomials are classified based on their degree and the number of terms. The question specifically deals with the degree of a single term, which is a monomial.
Classification by Degree:
- Degree 0: Constant polynomial (e.g., 5)
- Degree 1: Linear polynomial (e.g., \(2x + 1\))
- Degree 2: Quadratic polynomial (e.g., \(x^2 - 3x + 4\))
- Degree 3: Cubic polynomial (e.g., \(x^3 + 2\))
- Degree 4: Quartic polynomial (e.g., \(x^4 - x\))
Classification by Number of Terms:
- 1 term: Monomial (e.g., \(5x^3\))
- 2 terms: Binomial (e.g., \(2x + 1\))
- 3 terms: Trinomial (e.g., \(x^2 + 3x - 5\))
The problem focused only on calculating the degree of a single term by summing its variable exponents.
Important Questions from Polynomials
In the expansion of (x + 3) 3, the coefficient of x is:
The factorisation of x 2+ 11xy + 24y 2is:
The value of 16x 4+ 25y 2– 40x 2y at x = 5 and y = 2 is:
If the sum of the squares of the zeros of quadratic polynomial f(x) = x 2– 8x + k is 40, then find the value of k.
Expand : (s + 2) 3
A. s 3+ 2s 2+ 12s + 8
B. s 3+ 3s 2+ 6s + 8
C. s 3+ 6s 2+ 12s + 8
D. s 3+ 6s 2+ 6s + 8
