Question

Answer the following:
Write the degree of the polynomial $12 - 3x + 2{x^3}$.
Find the value of the polynomial $2{x^2} - 13{x^2} + 17x + 12$ at $x = 2$ .
Find the zeros of the polynomial $p\left( x \right) = x + 3$.

Answer 1

Hint: The given question consists of the three small questions based on the polynomials of different kinds. We will use the definitions of the concepts used in the question and then use it to solve the given problems. The three problems can be solved by using the definitions of the degree and zeros of the polynomials.

Complete step-by-step answer:
Observe that the three questions are based on the definitions of the basic concepts of the polynomials.
First question is related to the degree of the polynomial. For any polynomial $p\left( x \right)$ the degree of the polynomial is defined as the highest power of the variable $x$ in the polynomial.
For that purpose, we will first arrange the terms in the ascending order of the powers of the variables.
Thus, the given polynomial is written as $2{x^3} - 3x + 12$.
Now the highest power of $x$ is $3$, therefore, the degree of the given polynomial is $3$ .
Now the next question is about the value of the polynomial at a point.
We calculate the value of the polynomial at a specific point by replacing the variable of the polynomial by the given value for that polynomial.
In this case we will put $x = 2$ .
Assume that $p\left( x \right) = 2{x^3} - 3x + 12$ .
Therefore, $p\left( 2 \right) = 2{\left( 2 \right)^3} - 3\left( 2 \right) + 12$.
On simplifying we get, $p\left( 2 \right) = 22$.
Therefore, the value of the polynomial $12 - 3x + 2{x^3}$ at $x = 2$ is $22$ .
Last question is about the zeros of the polynomial.
The zeros of the polynomial are defined as the value of the variable that satisfies the given polynomial.
That means a real number $x = a$ is said to be zero of the polynomials $p\left( x \right)$ is $p\left( a \right) = 0$ .
We will start by putting the given polynomial equal to $0$ .
Therefore,
$x + 3 = 0$
Subtracting $3$ from both sides,
$x = - 3$
Therefore, zero of the polynomials $p\left( x \right) = x + 3$ is $x = - 3$ .


Note: First we started by expressing the given polynomials in the standard form where the term with highest power of the variable is the first and there after all the powers are in ascending order. We then used the definitions to reach the final answer. The problem consists of three simple questions, just you have to be careful while arranging the polynomial in the standard form so that we can directly use the definitions.