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Polynomials Test - 40

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Polynomials Test - 40
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  • Question 1
    1 / -0
    $$p(x) = 7x^2 - 9$$ is a ________  polynomial
    Solution
    The general form of quadratic polynomial is $$p(x)=ax^2+bx+c$$ where $$x$$ is a variable and $$a,b,c$$ are constants. The first term of this polynomial has power $$2$$ and the second term of this polynomial has power $$1$$ and therefore, the degree of the polynomial is the largest exponent that is $$2$$.

    Similarly, the polynomial $$p(x)=7x^2-9$$ has degree $$2$$.

    Hence, the polynomial $$p(x)=7x^2-9$$ is a quadratic polynomial.
  • Question 2
    1 / -0
    A polynomial of degree ________ is called a quadratic polynomial
    Solution
    The general form of quadratic polynomial is $$p(x)=ax^2+bx+c$$ where $$x$$ is a variable and $$a,b,c$$ are constants. The first term of this polynomial has power $$2$$ and the second term of this polynomial has power $$1$$ and therefore, the degree of the polynomial is the largest exponent that is $$2$$.

    Hence, a polynomial of degree $$2$$ is called a quadratic polynomial.
  • Question 3
    1 / -0
    $$p(x) = ax^2 + bx + c$$ is a ______.
    Solution
    The general form of quadratic polynomial is $$p(x)=ax^2+bx+c$$ where $$x$$ is a variable and $$a,b,c$$ are constants. 
    The degree of the polynomial is the largest exponent of $$x$$ that is $$2$$.

    Hence, the polynomial $$p(x)=ax^2+bx+c$$ is a quadratic polynomial.
  • Question 4
    1 / -0
    Which of the following is a quadratic polynomial?
    Solution
    $$\textbf{Step-1: Apply the concept of the polynomial.}$$

                     $$\text{The general form of quadratic polynomial is}$$ $$p(x)=ax^2+bx+c$$ 

                     $$\text{where}$$ $$x$$ $$\text{is a variable and}$$ $$a,b,c$$ $$\text{are constants}$$ ($$a\ne 0.$$) 

                     $$\text{The first term of this polynomial has power 2}$$  

                     $$\text{and the second term of this polynomial}$$ $$\text{ has power}$$ $$1$$ 

                     $$\text{and therefore, the degree of the polynomial is the largest exponent that is}$$ $$2$$.

                     $$\text{Similarly, the polynomial}$$ $$p(x)=5x^2-25$$ $$\text{has degree}$$ $$2$$.

                     $$\text{So, the polynomial}$$ $$p(x)=5x^2-25$$ $$\text{is a quadratic polynomial.}$$

    $$\textbf{Hence, correct option is C}$$
  • Question 5
    1 / -0
    Which of the following is not a linear polynomial?
    Solution
    $$\textbf{Step1: Check Every Option to Find whether it is Linear Polynomial or Not}$$
                   $$\text{The General form of linear polynomial is p(x)=ax+b}$$
                   $$\text{where x is a Variable and b is a Constant}$$ 
                   $$\text{The first term of this polynomial has power 1 and Therefore, the degree of the polynomial is 1.}$$
                   $$\text{Since Linear Polynomial is Polynomial in terms Single variable with degree 1}$$ 
                   $$\text{ By Verifying every option,we can say that-}$$
                   $$\text{Option A.}$$ $$p(y)=8y+6$$ $$\text{which is a Linear Polynomial with Degree 1}$$ 
                   $$\text{Option B.}$$   $$p(x)=8y+2x$$ $$\text{which is a Linear Polynomial in 2 Variable.}$$  
                   $$\text{Option C.}$$  $$p(x)=4+\dfrac { 5x }{ x } +8x^{ 0 }$$ $$=4+5+1$$ $$\text{=10 is a Constant Term.}$$
                   $$\text{Option D.}$$ $$p(y)=4+5x$$ $$\text{which is a Linear Polynomial with Degree 1.}$$    

    $$\textbf{Hence, the Polynomial $p(x)=4+\dfrac { 5x }{ x } +8x^{ 0 }$ is not a Linear Polynomial.}$$
                                                       

  • Question 6
    1 / -0
    Which of the following is not a linear polynomial?
    Solution
    $$\textbf{Step1: Check Every Option to Find whether it is Linear Polynomial or Not}$$
                   $$\text{The General form of linear polynomial is p(x)=ax+b}$$
                   $$\text{where x is a Variable and b is a Constant}$$ 
                   $$\text{The first term of this polynomial has power 1 and Therefore, the degree of the polynomial is 1.}$$
                   $$\text{Since Linear Polynomial is Polynomial in terms Single variable with degree 1}$$ 
                   $$\text{By Verifying every option,we can say that-}$$
                   $$\text{Option A.}$$  $$p(y)=8y^{-1}+4$$ $$\text{It is not a Linear Polynomial since Power of y is not a Whole Number.}$$ 
                   $$\text{Option B.}$$   $$p(x)=8x+4$$ $$\text{which is a Linear Polynomial with Degree 1.}$$      
                   $$\text{Option C.}$$  $$p(y)=8y+4$$ $$\text{which is a Linear Polynomial with Degree 1.}$$    
                   $$\text{Option D.}$$  $$p(z)=8z+4 $$  $$\text{which is a Linear Polynomial with Degree 1.}$$    

    $$\textbf{Hence, the Expression $\boldsymbol{p(x)=8y^{ -1 }+4}$ is not a Linear Polynomial.}$$
  • Question 7
    1 / -0
    $$p(x) = x^3 - x^2 - x + x$$ is a _________.
    Solution
    The general form of cubic polynomial is $$p(x)=ax^3+bx^2+cx+d$$ where $$x$$ is a variable and $$a,b,c,d$$ are constants. The first term of this polynomial has power $$3$$, the second term of this polynomial has power $$2$$ and the third term of this polynomial has power $$1$$ and therefore, the degree of the polynomial is the largest exponent that is $$3$$.

    Similarly, in the polynomial $$p(x)=x^3-x^2-x+x$$ or $$p(x)=x^3-x^2$$, the first term of this polynomial has power $$3$$ and the second term of this polynomial has power $$2$$ and therefore, the degree of the polynomial is the largest exponent that is $$3$$.

    Hence, the polynomial $$p(x)=x^3-x^2-x+x$$ is a cubic polynomial.
  • Question 8
    1 / -0
    Which of the following is NOT a quadratic polynomial?
    Solution
    $$\textbf{Step1: Check Every Option to Find weather it is Quadratic Polynomial or Not}$$
                    $$\text{The General Form of Quadratic Polynomial is p(x)}$$ $$=ax^2+bx+c$$
                    $$\text{where x is a Variable and a,b,c are constants}$$    

                    $$\text{The First term of this Polynomial has Power 2 and the Second term has Power 1}$$
                     $$\therefore\text{Degree of the Polynomial is the Largest Exponent that is 2}$$ 

                   $$\text{ By Verifying every option,we can say that-}$$

                   $$\text{Option A.}$$  $$p(x)=px^{2}+qx+r$$ $$\text{which is a Quadratic Polynomial with Degree 2.}$$      

                   $$\text{Option B.}$$   $$p(x)$$$$=ax^2+bx+c$$$$\text{which is a Quadratic Polynomial with Degree 2.}$$   
                
                   $$\text{Option C.}$$$$p(x)=x.x^2+x.x+x$$$$=x^3+x^2+x$$$$\text{ which is a not a Quadratic Polynomial.}$$  
                   $$\text{Since Degree i.e. Highest Power of Polynomial is 3.}$$        
                         
                   $$\text{Option D.}$$  $$p(x)=$$$$=mx^2+nx+1$$$$\text{which is a Quadratic Polynomial with Degree 2.}$$    
       
    $$\textbf{Hence, the polynomial $p(x)=x.x^2+x.x+x$ is not a Quadratic polynomial.}$$


  • Question 9
    1 / -0
    $$p(x) = 2x^3 - 3x^2 - 5$$ is a ________ polynomial.
    Solution
    The general form of cubic polynomial is $$p(x)=ax^3+bx^2+cx+d$$ where $$x$$ is a variable and $$a,b,c,d$$ are constants. The first term of this polynomial has power $$3$$, the second term of this polynomial has power $$2$$ and the third term of this polynomial has power $$1$$ and therefore, the degree of the polynomial is the largest exponent that is $$3$$.

    Similarly, in the polynomial $$p(x)=2x^3-3x^2-5$$, the first term of this polynomial has power $$3$$ and the second term of this polynomial has power $$2$$ and therefore, the degree of the polynomial is the largest exponent that is $$3$$.

    Hence, the polynomial $$p(x)=2x^3-3x^2-5$$ is a cubic polynomial.
  • Question 10
    1 / -0
    The constant polynomial whose coefficients are all equal to $$0$$ is called ________ polynomial.
    Solution
    For example:-
    Consider the polynomial, $$p(x) = ax^2 + bx +c$$ , if $$a=b=c= 0$$ then the expression becomes zero polynomial. 

    Therefore, zero polynomial can be written as $$p(x) = 0$$. 

    Hence, the constant polynomial whose coefficients are all equal to $$0$$ is called a zero polynomial. 
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