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Theory:

The value of the polynomial $$p(x)$$ at $$x=a$$ is $$p(a)$$ acquired when $$x$$ is replaced by $$a$$ ($$a∈R$$).
Example:
Write the value of $$p(x) = x^2+2x-1$$ at $$x = 3$$.

The value of $$p(x)$$ at $$x = 3$$ can be obtained by substituting the point $$x = 3$$ in the polynomial.

Substitute $$x=3$$ in the polynomial $$p(x)$$.

$$p(3) =$$$$3^2+2(3)-1$$

$$=9+6-1$$

$$=14$$.
Zero of polynomial $$p(x)$$ is a real number '$$a$$' such that $$p(a)= 0$$.
For polynomial $$p(x)$$, if $$p(a) = 0$$ then the zero polynomial is $$x = a$$.

So, we put $$p(x) = 0$$ to find zero polynomial.

The value of $$x$$ is then found.

Important!
1. Zero of a polynomial: To find the zero of polynomial, we put $$p(x) =$$ $$0$$ in the given polynomial $$p(x)$$.
Example:
i. Consider $$p(x)=3a$$

Putting $$p(x) = 0$$ in the polynomial equation.

Thus, $$a=0$$ is the zero of the polynomial $$p(a)=3a$$.

ii. Consider the polynomial $p\left(x\right)=x-2$.

Putting $$p(x) = 0$$ in the polynomial equation.

$\begin{array}{l}0=x-2\\ \\ x=2\end{array}$

Thus $$x=2$$ is the zero of the polynomial $p\left(x\right)=x-2$.
2. Zeros of a polynomial $p\left(x\right)$ is the real number '$a$' for which $p\left(x\right)$ if $p\left(a\right)=0$. In this situation, we say $$p(x) = 0$$ is a polynomial equation, and $a$ is a root of the polynomial.
• Each real number is a zero of the zero polynomial $p\left(x\right)=0$.
Example:
The polynomial $$p(8) = 0$$, where $$8$$ is a real number.
• The non -zero constant polynomial does not have zeros.
Example:
The polynomial $p\left(x\right)=8$ does not have a Zero.