Concept explainers
Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial as constant, linear, quadratic, cubic, or quartic.
1. f(x)=2x3+6x2−x4+11
To find:
The leading term, the leading coefficient, the degree and the type of given polynomial f(x)=2x3+6x2−x4+11.
Answer to Problem 1CT
Solution:
The leading term is −x4. The leading coefficient is −1. The degree of the polynomial is 4. The type of the polynomial is quartic.
Explanation of Solution
Given:
The given polynomial is f(x)=2x3+6x2−x4+11.
Concept:
(a) The leading term is the first term of a polynomial arranged in descending order of their powers.
(b) The leading coefficient is the coefficient of leading term.
(c) The degree is the highest power of a polynomial.
(d) The type of a polynomial is defined by its degree.
1 A polynomial with degree zero is the constant polynomial function.
2 A polynomial with degree one is the linear polynomial function.
3 A polynomial with degree two is the quadratic polynomial function.
4 A polynomial with degree three is the cubic polynomial function.
5 A polynomial with degree four is the quartic polynomial function.
The given polynomial f(x) is in descending order,
f(x)=−x4+2x3+6x2+11.
The first term of the polynomial is −x4. So, the leading term is −x4.
The coefficient of the leading term is −1. So, the leading coefficient is −1.
The leading term has the highest power of 4. So, the degree of the polynomial is 4.
The degree of the polynomial is 4. So, the polynomial is a quartic polynomial.
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