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Define zero or root of a polynomial.

seo-qna
Last updated date: 27th Mar 2024
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MVSAT 2024
Answer
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Hint: First of all, try to recollect what polynomial is. Now try to recollect the value of the variable which makes this polynomial zero like in polynomial 2x – 4 = 0, the value of variable x = 2 is making the polynomial 0. Therefore, define the zero or root of the polynomial.

Complete step-by-step solution -
In this question, we have to define the zero or root of the polynomial. Let us first see what polynomials are.
Polynomial: A polynomial is defined as an expression that contains two or more algebraic terms. It is made up of two terms namely Poly (meaning “many”) and Nominal (meaning “terms.”). Polynomials are composed of:
Constants such as 1, 2, 3, etc.
Variables such as g, h, x, y, etc.
Exponents such as 5 in \[{{x}^{5}}\]etc.
The polynomial function is denoted by P(x) where x represents the variable. For example,
\[P\left( x \right)={{x}^{2}}-10x+15\]
Now let us define what zero or root of a polynomial is.
Zeros or roots of a polynomial: For a polynomial, there could be some values of the variable for which the polynomial will be zero. These values are called zeros of a polynomial. We can also call these as roots of the polynomials. To find the zeroes of a polynomial, we equate it to zero and find the value of the variable. For example if we have polynomial as P(x) = 4x + 3. We find its zero or root as follows:
\[P\left( x \right)=4x+3=0\]
\[\Rightarrow 4x=-3\]
\[\Rightarrow x=\dfrac{-3}{4}\]
So, we get the zero or root of polynomial P(x) = 4x+3 as \[\dfrac{-3}{4}.\]

Note: Students must note that we don’t have a proper method of finding the zeros of higher degree polynomials but we can always check if a number is a zero of a polynomial or not by substituting that value of zero in place of the variable in that polynomial and checking if that expression becomes zero or not. For example, here we have to check that \[x=\dfrac{-3}{4}\] is a root of P(x) = 4x + 3 or not. We will do it as follows:
\[P\left( x \right)=4x+3\]
By substituting \[x=\dfrac{-3}{4}\], we get,
\[\Rightarrow P\left( x \right)=4\times \left( \dfrac{-3}{4} \right)+3\]
\[\Rightarrow P\left( x \right)=\dfrac{-12}{4}+3\]
\[\Rightarrow P\left( x \right)=-3+3\]
\[\Rightarrow P\left( x \right)=0\]
So, \[x=\dfrac{-3}{4}\] is a root or zero of P(x) = 4x + 3.

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