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Hint: The zero of the polynomial is that number which when substituted in the polynomial gives 0 . Thus, to check whether the given number is the zero of the polynomial, substitute $x = - \dfrac{1}{3}$ in the given polynomial. If the result is 0, then the answer is yes otherwise not a 0 of the polynomial.

The maximum number of zeroes for a polynomial is determined by the highest degree of the polynomial.

The highest degree of a polynomial is the maximum power that the variable is raised to in the polynomial.

We are given that the polynomial is $p\left( x \right) = 3x + 1$.

In the given polynomial $p\left( x \right) = 3x + 1$, the highest power that the variable is raised to is 1. Therefore, the degree of the given polynomial is 1.

Therefore, the given polynomial has at most 1 zero.

We want to find out whether $x = - \dfrac{1}{3}$ is the zero of the given polynomial or not.

As we know, the zero of the polynomial is that number which when substituted in the polynomial gives the answer 0.

Now, we will substitute $x = - \dfrac{1}{3}$ in the polynomial $p\left( x \right) = 3x + 1$

On substituting, we get,

$

p\left( { - \dfrac{1}{3}} \right) = 3\left( { - \dfrac{1}{3}} \right) + 1 \\

p\left( { - \dfrac{1}{3}} \right) = - 1 + 1 \\

p\left( { - \dfrac{1}{3}} \right) = 0 \\

$

From the above result we can conclude that $x = - \dfrac{1}{3}$ is the zero of the polynomial.

Note: The maximum number of zeroes for a polynomial is determined by the highest degree of the polynomial. The highest degree of a polynomial is the maximum power that the variable is raised to in the polynomial. In linear polynomials, we can calculate the zero of the polynomial by putting it to 0 and then finding the value of, which will be the zero of the polynomial.

__Complete step by step answer:__The maximum number of zeroes for a polynomial is determined by the highest degree of the polynomial.

The highest degree of a polynomial is the maximum power that the variable is raised to in the polynomial.

We are given that the polynomial is $p\left( x \right) = 3x + 1$.

In the given polynomial $p\left( x \right) = 3x + 1$, the highest power that the variable is raised to is 1. Therefore, the degree of the given polynomial is 1.

Therefore, the given polynomial has at most 1 zero.

We want to find out whether $x = - \dfrac{1}{3}$ is the zero of the given polynomial or not.

As we know, the zero of the polynomial is that number which when substituted in the polynomial gives the answer 0.

Now, we will substitute $x = - \dfrac{1}{3}$ in the polynomial $p\left( x \right) = 3x + 1$

On substituting, we get,

$

p\left( { - \dfrac{1}{3}} \right) = 3\left( { - \dfrac{1}{3}} \right) + 1 \\

p\left( { - \dfrac{1}{3}} \right) = - 1 + 1 \\

p\left( { - \dfrac{1}{3}} \right) = 0 \\

$

From the above result we can conclude that $x = - \dfrac{1}{3}$ is the zero of the polynomial.

Note: The maximum number of zeroes for a polynomial is determined by the highest degree of the polynomial. The highest degree of a polynomial is the maximum power that the variable is raised to in the polynomial. In linear polynomials, we can calculate the zero of the polynomial by putting it to 0 and then finding the value of, which will be the zero of the polynomial.

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