Answer the questions about the following polynomial:
$-\frac{x^4}{9}+5$
The expression represents a polynomial with $\square$ terms. The constant term is $\square$, and the leading coefficient is $\square$. The leading term is $\square$.
Answer (2)
The expression is a polynomial.
The polynomial has 2 terms.
The leading term is − 9 x 4 .
The constant term is 5 , and the leading coefficient is − 9 1 .
The final answer is: The expression represents a polynomial with 2 terms. The leading term is − 9 x 4 , the constant term is 5 , and the leading coefficient is − 9 1 .
Explanation
Understanding the Polynomial The given expression is − 9 x 4 + 5 . We need to identify the type of expression, the number of terms, the leading term, the constant term, and the leading coefficient.
Identifying the Type of Expression The expression is a polynomial because it consists of terms with non-negative integer exponents of the variable x .
Counting the Terms The polynomial has two terms: − 9 x 4 and 5 .
Finding the Leading Term The leading term is the term with the highest power of x , which is − 9 x 4 .
Identifying the Constant Term The constant term is the term without any variable, which is 5 .
Determining the Leading Coefficient The leading coefficient is the coefficient of the leading term, which is − 9 1 .
Final Answer Therefore, the expression represents a polynomial with 2 terms. The leading term is − 9 x 4 , the constant term is 5 , and the leading coefficient is − 9 1 .
Examples
Polynomials are used in various real-world applications, such as modeling curves and shapes in computer graphics, describing the trajectory of a projectile in physics, and representing relationships in economics and finance. For example, the path of a ball thrown in the air can be approximated by a quadratic polynomial, allowing us to predict its landing point. Similarly, polynomials can be used to model the growth of a population or the decay of a radioactive substance.
The given expression − 9 x 4 + 5 represents a polynomial with 2 terms. The leading term is − 9 x 4 , the constant term is 5 , and the leading coefficient is (-\frac{1}{9}.
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