Question

The maximum number of terms in a polynomial of degree 10 is:
(a). 9
(b). 10
(c). 11
(d). 1

Answer 1

Hint:We should know that all the algebraic expressions with whole numbers as the exponents of the variable are called polynomials.
The highest power (or exponent) of a variable in the polynomial is called degree.
A polynomial of degree one is called linear polynomial. E.g.: $3x$.
A polynomial of degree two is called a quadratic polynomial. E.g.: $2y + 7$.
A polynomial of degree three is called a cubic polynomial. E.g.: $\mathop z\nolimits^2 + 3z + 5$.

Complete step-by-step answer:
A polynomial \[p\left( x \right)\]in one variable x is an algebraic expression in x of the form
$p\left( x \right) = \mathop a\nolimits_n \mathop x\nolimits^n + \mathop a\nolimits_{n - 1} \mathop x\nolimits^{n - 1} + ... + \mathop a\nolimits_2 \mathop x\nolimits^2 + \mathop a\nolimits_1 x + \mathop a\nolimits_0 $
Where \[\mathop a\nolimits_0 ,\mathop {{\text{ }}a}\nolimits_1 ,\mathop {{\text{ }}a}\nolimits_2 ,...\mathop {,{\text{ }}a}\nolimits_n \]are constants and \[\mathop a\nolimits_n \ne 0\].
\[\mathop a\nolimits_0 ,\mathop {{\text{ }}a}\nolimits_1 ,\mathop {{\text{ }}a}\nolimits_2 ,...\mathop {,{\text{ }}a}\nolimits_n \]are respectively coefficients of $\mathop x\nolimits^0 ,\mathop x\nolimits^1 ,\mathop x\nolimits^2 ,...,\mathop x\nolimits^n $, and
$n$is called the degree of the polynomial.
Each of $\mathop a\nolimits_n \mathop x\nolimits^n ,\mathop a\nolimits_{n - 1} \mathop x\nolimits^{n - 1} ,...,\mathop a\nolimits_2 \mathop x\nolimits^2 ,\mathop a\nolimits_1 x,\mathop a\nolimits_0 $ with \[\mathop a\nolimits_n \ne 0\] is called a term of the polynomial $p\left( x \right)$.
Step 2: It is given to find the number of terms in the polynomial of degree 10.
This implies degree $n = 10$ in polynomial $p\left( x \right)$.
Step 3: Substitute the value of $n = 10$ in polynomial $p\left( x \right)$
$p\left( x \right) = \mathop a\nolimits_{10} \mathop x\nolimits^{10} + \mathop a\nolimits_9 \mathop x\nolimits^9 + \mathop a\nolimits_8 \mathop x\nolimits^8 + \mathop a\nolimits_7 \mathop x\nolimits^7 + \mathop a\nolimits_6 \mathop x\nolimits^6 + \mathop a\nolimits_5 \mathop x\nolimits^5 + \mathop a\nolimits_4 \mathop x\nolimits^4 + \mathop a\nolimits_3 \mathop x\nolimits^3 + \mathop a\nolimits_2 \mathop x\nolimits^2 + \mathop a\nolimits_1 x + \mathop a\nolimits_0 $
Where \[\mathop a\nolimits_0 ,{\text{ }}\mathop a\nolimits_1 ,{\text{ }}\mathop a\nolimits_2 ,{\text{ }}\mathop a\nolimits_3 ,{\text{ }}\mathop a\nolimits_4 ,{\text{ }}\mathop a\nolimits_5 ,{\text{ }}\mathop a\nolimits_6 ,\mathop {{\text{ }}a}\nolimits_7 ,{\text{ }}\mathop a\nolimits_8 ,{\text{ }}\mathop {\mathop a\nolimits_9 ,{\text{ }}a}\nolimits_{10} \] are constants and \[\mathop a\nolimits_n \ne 0\].
\[\mathop a\nolimits_0 ,{\text{ }}\mathop a\nolimits_1 ,{\text{ }}\mathop a\nolimits_2 ,{\text{ }}\mathop a\nolimits_3 ,{\text{ }}\mathop a\nolimits_4 ,{\text{ }}\mathop a\nolimits_5 ,{\text{ }}\mathop a\nolimits_6 ,\mathop {{\text{ }}a}\nolimits_7 ,{\text{ }}\mathop a\nolimits_8 ,{\text{ }}\mathop {\mathop a\nolimits_9 ,{\text{ }}a}\nolimits_{10} \] are respectively coefficients of $\mathop x\nolimits^0 ,\mathop x\nolimits^1 ,\mathop x\nolimits^2 ,\mathop x\nolimits^3 ,\mathop x\nolimits^4 ,\mathop x\nolimits^5 ,\mathop x\nolimits^6 ,\mathop x\nolimits^7 ,\mathop x\nolimits^8 ,\mathop x\nolimits^9 ,\mathop x\nolimits^{10} $.
Each of \[\mathop a\nolimits_{10} \mathop x\nolimits^{10} ,{\text{ }}\mathop a\nolimits_9 \mathop x\nolimits^9 ,\mathop {{\text{ }}a}\nolimits_8 \mathop x\nolimits^8 ,{\text{ }}\mathop a\nolimits_7 \mathop x\nolimits^7 ,{\text{ }}\mathop a\nolimits_6 \mathop x\nolimits^6 ,{\text{ }}\mathop a\nolimits_5 \mathop x\nolimits^5 ,\;\mathop a\nolimits_4 \mathop x\nolimits^4 ,\;\mathop a\nolimits_3 \mathop x\nolimits^3 ,\;\mathop a\nolimits_2 \mathop x\nolimits^2 ,\;\mathop a\nolimits_1 x,\;\mathop a\nolimits_0 \] are the terms of the polynomial $p\left( x \right)$.
Step 4: Count the numbers of terms in polynomial $p\left( x \right)$
Numbers of terms = 11
Final answer: There are 11 terms in the polynomial of degree 10. The correct option is (C).

Note: Following results can be concluded from the above solution.
Numbers of terms in the polynomial of degree $n = 1$is 2.
The Number of terms in the polynomial of degree $n = 2$is 3.
Therefore, the number of terms in the polynomial of degree $n$is $n + 1$.
A polynomial can have any (finite) numbers of terms. For instance, $\mathop x\nolimits^{150} + \mathop x\nolimits^{149} + ... + \mathop x\nolimits^2 + \mathop a\nolimits_1 x + \mathop a\nolimits_0 $ is a polynomial with 151 terms.
The degree of the non-zero polynomial is 0. For example, degree of polynomial,
$p\left( x \right)$= 7, degree is 0.