Answer

Verified

429.6k+ views

**Hint:**

We are asked to find the value of k such that 3 is zero of the polynomial \[2{{x}^{2}}+x+k.\] We will first learn the definition of zero of a polynomial and then we will consider that x = 3 is a zero and compare P(3) = 0 to find the value of k where \[P\left( x \right)=2{{x}^{2}}+x+k.\]

**Complete step by step answer:**

We are given a polynomial \[2{{x}^{2}}+x+k\] and mentioned that we have to find such a value of k so that 3 is a root of the given above polynomial. To solve for k, we will first learn about the roots (zeroes). Zeroes (roots) are those values of x which when inserted in the place of x in the polynomial P(x) then the result will be zero such that P(x) = 0 are called zeros of the polynomial.

For example if we take P(x) = x + 2 that is at x = – 2. We have,

\[P\left( -2 \right)=-2+2=0\]

So, – 2 is zero of P(x) = x + 2. Similarly, let \[P\left( x \right)={{x}^{2}}-1\] so x = 1 is zero of P(x).

Now, we are having a polynomial \[2{{x}^{2}}+x+k\] we are asked to find k such that 3 is zero of the polynomial.

Let x = 3 is zero of \[P\left( x \right)=2{{x}^{2}}+x+k.\] So by definition of zero, we put x = 3, we get P(3) = 0. So,

\[P\left( 3 \right)=2{{\left( 3 \right)}^{2}}+3+k=0\]

Now, we simplify,

\[\Rightarrow 2\times 9+3+k=0\left[ \text{As }{{3}^{2}}=9 \right]\]

\[\Rightarrow 18+3+k=0\]

So solving, we get,

\[\Rightarrow 21+k=0\]

This implies we have,

\[\Rightarrow k=-21\]

Hence, we got that for k = – 21, x = 3 is a zero of the polynomial, \[2{{x}^{2}}+x+k.\]

We can cross-check as below. We will put k = – 21 in \[2{{x}^{2}}+x+k\] so we have

\[P\left( x \right)=2{{x}^{2}}+x-21\]

Now we will find its zero. So, by splitting the middle term, we have,

\[\Rightarrow 2{{x}^{2}}+x-21=2{{x}^{2}}+7x-6x-21\]

On simplifying, we get,

\[\Rightarrow 2{{x}^{2}}+x-21=x\left( 2x+7 \right)-3\left( 2x+7 \right)\]

On further solving, we get,

\[\Rightarrow 2{{x}^{2}}+x-21=\left( x-3 \right)\left( 2x+7 \right)\]

So, we get,

\[\Rightarrow x-3=0;2x+7-0\]

\[\Rightarrow x=-3;x=\dfrac{-7}{2}\]

So, the zeroes are 3 and \[\dfrac{-7}{2}.\]

Hence, 3 is a zero, so we get k as – 21.

**Note:**

There is an easy way to check this. We have \[2{{x}^{2}}+x-21,\] we have to check if 3 is a zero of this, so we put x = 3 and check we get 0 as an answer or not. So, we get,

\[P\left( 3 \right)=2{{\left( 3 \right)}^{2}}+3-21\]

Simplifying, we get,

\[P\left( 3 \right)=2\times 9+3-21\]

\[\Rightarrow P\left( 3 \right)=18+3-21\]

\[\Rightarrow P\left( 3 \right)=21-21\]

\[\Rightarrow P\left( 3 \right)=0\]

As we get P(3) = 0, so k = – 21 is the correct solution.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is BLO What is the full form of BLO class 8 social science CBSE