Most Important Questions for Class 10 Math Ch 2 Polynomials with Solutions

Polynomials are an important chapter of mathematics; they form a fundamental basis of the curriculum. Understanding polynomials is important as they provide a base for more advanced concepts that students may face in their future studies. In Class 10, chapter 2 of the math is named Polynomials.

To help students excel in their exams, here we will explore the most important questions, class 10 maths ch 2 extra questions, to ensure that our students are always equipped with the right study materials and have a good understanding of the chapter.

Constants: 1, 5, 8, etc

Variables: x, y, p, r, etc 

Exponents: 6 in x⁵, etc.

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Chapter 2 Polynomials: Important Questions

1. (x²-3√2x+4)/(x²-√2)   ; x =√2

At how many points does the graph of the above expression intersect the x-axis? Show your work.

Soln. 

The points where the graph of the above expression intersects the x-axis represent the zero of a polynomial.

The zero of a polynomial is obtained by equating fx to zero.

fx=0

⇒x²-32x+4x-2=0

Now factorising the numerator, we get

⇒x²-2√2x-2x+4/(x-√2)=0

⇒xx-2√2-√2x-2√2/(x-√2)=0

⇒x-√2x-2√2x-√2=0

So, we get x-2√2=0

Hence, there is only one point where the graph of the expression intersects the x-axis is x=2√2

2. p and q are zeroes of the polynomial 2x²+5x-4.

Without finding the actual values of p and q, evaluate (1 - p)(1 − q ). Show your steps.

Soln. 

‍Given that, p and q are the zeroes of a polynomial 2x²+5x-4.

We need to find the value of  (1-p)(1-q).

(1-p)(1-q)=1-q-p+pq

(1-p)(1-q)=1-p+q+pq ...(i)

Using the relation between the zeroes and the coefficients of the quadratic polynomial, we get

Sum of the zeroes =-b/a

Product of the zeroes =c/a

where, the standard form of a quadratic polynomial is fx=ax2+bx+c

Here in the given polynomial, we get a=2, b=5, c=-4

So, the sum of the zeroes =-5/2

The product of the zeroes =-4/2=-2

So, p+q=-5/2 and pq=-2

Substituting the value of p+q and pq in equation(i), we get

(1-p1-q)=1-(-5/2)+(-2)

=-1+5/2=3/2

Hence, the value of (1-p)(1-q) is 3/2

3. The sum of two zeroes of q ( x ) is zero.Using the relationship between the zeroes and coefficients of a polynomial, find the:i) zeroes of q ( x ).ii) value of k.Show your steps.

A polynomial is given by q ( x ) = x³- 2x²-9x+k, where k is a constant.

Soln. 

q ( x ) = x³- 2x²-9x+k

α+β=0

This implies:

β=−α

From α+β+γ=2

= 2α+β+γ=2 and α+β=0

γ=2

Substitute β=−α and γ=2 into αβ+βγ+γα=−9:

α(−α)+(−α)(2)+(2)(α)=−9

Simplify:

−α²−2α+2α=−9

−α²=−9

α²=9

Thus:

α=3 orα=−3

If α=3 then β=−3 then If α=-3 then β=3

From αβγ=−k, substitute α=3, β=−3, and γ=2:

(3)(−3)(2)=−k

-18 = -k

k=18

Verify the Zeroes

The zeroes are α=3, β=−3, and γ=2. Verify:

1. Sum of the zeroes: 3+(−3)+2=2 (correct).
2. Product of zeroes taken two at a time:3(−3)+(−3)(2)+(2)(3)=−9 (correct).
3. Product of zeroes: 3(−3)(2)=−18 so k=18 (correct).

Final Answer:

i) The zeroes of q(x) are 3, −3, and 2.
ii) The value of k is 18.

4. p ( x ) = ax² - 8 x + 3, where a is a non-zero real number. One zero of p ( x ) is 3 times the other zero.

i) Find the value of a. Show your work.

ii) What is the shape of the graph of p (x)?

Give a reason for your answer.

Soln. 

Let the two zeroes of p(x) be α and β. It is given that:

β=3α

From the relationships between zeroes and coefficients of the polynomial ax² - 8 x + 3:

Sum of the zeroes: α+β

=-(-8/a)

=8/a

Product of the zeroes: αβ

=3/a

Substitute β=3α into the sum of the zeroes:

α+3α=8/a​

4α=8/a​

α=2/a​

Substitute α=2\α and β=3α=6/a into the product of the zeroes:

αβ= (2/a) x (6/a)

= 12/a²

Equating to the product relationship:

12/a² = 3/a

(since a≠0):

12=3a 

a=4

5. p(x) = 2x²-6x-3. The two zeroes are of the form:

(3 ±√k)/2; Where k is a real numberUse the relationship between the zeroes and coefficients of a polynomial to find the value of k. Show your steps.

Soln. 

‍p(x)=2x²−6x−3

The roots of p(x) are given in the form:

x_1​,x_2​= (3±√k)/2

Here, k is an unknown value that we need to determine.

Sum of the zeroes: x_1​+ x₂

=- (-6/2)

= 3

Product of the zeroes: x_1​. x₂

=- (3/2)

= -3/2

From the roots x_1​. x₂= (3±√k)/2, compute the sum of the zeroes:

Sum of the zeroes: x_1​+ x₂

= (3+√k)/2 +(3-√k)/2

= (3+√k +(3-√k)/2

= 6/2

= 3

Product of the zeroes: x_1​. x₂

=(3+√k)/2 . (3-√k)/2

= (3²-√k²)/2²

= (9-k)/4

Equating this to the given product of the zeroes

(9-k)/4 = -3/2

9−k=−6

k=9+6=15

k=15

6. Find the distance between the zeroes of the polynomial f ( x ) = 2 x²-x-6. Show your steps.

Soln. For f ( x ) = 2 x²-x-6 the coefficients are:

a=2,

b=−1,

c=−6.

The quadratic formula gives the zeroes of the quadratic equation:x= (−b±(√b²-4ac))/2a

x= (−(-1)±(√(-1)²-4(2)(-6)))/2(2)

x= (1±7​)/4

Thus, the two zeroes are:

x₁​= (1+7​)/4

x₁​= (8​)/4

x₁​= 2

x₂​= (1-7​)/4

x₂= (-6​)/4

x₂​= (-3/2)

Distance=∣x₁​−x₂​∣

Substitute the values of x₁​= 2 and x₂​= (-3/2)

Distance=7​/2

Distance= 3.5

7. Shown below are the graphs of two cubic polynomials, f ( x ) and g (x). Both polynomials have the zeroes (-1), 0 and 1.

Anya said, "Both the graphs represent the same polynomial, f ( x ) = g(x) = (x + 1)( x-0)(x-1) 

as they have the exact zeroes." Pranit said, "Both the graphs represent two different polynomials, f ( x ) = (x + 1)(x- 0)(x-1) and g(x) = (x + 1)(x - 0)(x-1) and only two such polynomials exist that can have the zeroes (-1), 0 and 1."

Aadar said, "Both the graphs represent two different polynomials, and infinitely many such polynomials exist that have the zeroes (-1), 0 and 1." Who is right? Justify your answer.

Soln. 

To determine who is correct among Anya, Pranit, and Aadar, let us analyze the situation:

Given:

• Both graphs have the same zeroes, i.e., x=−1, x=0, and x=1.
• The polynomial expressed as (x+1)(x−0)(x−1) has these zeroes.

Analysis:

Anya's Claim: Anya said, "Both the graphs represent the same polynomial f(x)=g(x)=(x+1)(x)(x−1) because they have the exact same zeroes."

Reason Anya is incorrect:

• The zeroes only indicate the locations where the graphs intersect the x-axis, but the shape of the graphs depends on the leading coefficient of the polynomial.
• Two polynomials can share the same zeroes but differ in their leading coefficients or scaling factors. For example:

f(x)=(x+1)(x)(x−1)

g(x)=c(x+1)(x)(x−1), where c is a constant other than 1.

• In this case, the graphs of f(x) and g(x) differ in their height (vertical scaling).

Pranit's Claim: Pranit said, "Both the graphs represent two different polynomials, f(x)=(x+1)(x)(x−1) and g(x)=(x+1)(x)(x−1) and only two such polynomials exist that can have the zeroes −1, 0, and 1."

Reason Pranit is incorrect:

• His statement is contradictory. If both graphs represent different polynomials, there can be infinitely many such polynomials with the same zeroes because the general form of a polynomial with zeroes −1, 0, and 1 is:

P(x)=c(x+1)(x)(x−1)

Here, ccc can be any real number (not just 1 or some fixed constant).

Aadar's Claim: Aadar said, "Both the graphs represent two different polynomials, and infinitely many such polynomials exist that have the zeroes −1, 0, and 1."

Reason Aadar is correct:

• A polynomial with roots −1, 0, and 1 has the general form

P(x)=c(x+1)(x)(x−1)

• The constant c determines the vertical scaling of the polynomial. For example

If c=1, the polynomial is f(x)=(x+1)(x)(x−1)

If c=2, the polynomial is g(x)=2(x+1)(x)(x−1)

This means there are infinitely many possible polynomials (corresponding to different values of c) with the same zeroes.

Aadar is correct.

The two graphs represent different polynomials with the same zeroes, and there are infinitely many such polynomials because any polynomial of the form P(x)=c(x+1)(x)(x−1) where c is any non-zero constant, will have the zeroes −1, 0, and 1.

8. p(x) = (x+3)² - 2(x- c ); where c is a constant.

If p (x) is divisible by x, find the value of c. Show your steps.

Soln. 

Given a polynomial px=x+3²-2x-c, where c is a constant.

Using algebraic identity, we can expand the expression,

px=x²+6x+9-2x+2c

px=x²+4x+9+2c

We need to find the value of c when px is divisible by x

According to the remainder theorem, when a polynomial p(x) (whose degree is greater than or equal to 1 is divided by a linear polynomial qx) whose zero is x=a, the remainder is given by r=pa

Substituting x=0, we get

Remainder =p0=9+2c

Since px is divisible by x, we get the remainder as zero.

⇒9+2c=0

2c=-9

c=-9/2

Hence, the value of c is -9/2

9. Students of a class were shown the graph below.

Based on their answers, they were divided into two groups. Group 1 said the graph represented a quadratic polynomial, whereas group 2 said the graph represented a cubic polynomial.

i) Which group was correct?

ii) Write the polynomial represented by the graph.

Soln. 

(i) The graph clearly shows three x-intercepts (roots), indicating that the polynomial has degree 3. Therefore, the graph represents a cubic polynomial. Thus, Group 2 is correct.

(ii) From the graph, the x-intercepts are at x=−2, x=0, and x=2.

If the roots are −2, 0, and 2, the polynomial can be expressed as:

p(x)=k(x+2)(x)(x−2)

Here, k is a constant to be determined.

Using the difference of squares:

p(x)=k(x²−4)x

p(x)=k(x³−4)

From the graph, the maximum value of p(x) appears to be approximately 10 at x=−1.

Substitute x=−1 and p(x)=10 into the polynomial:

10=k((−1)³−4(−1)) 

10=k(−1+4)

10=k(3)

k= 10/3

Substituting k= 10/3 into the equation

p(x)= 10/3(x³−4x)

Thus, the polynomial represented by the graph is:

p(x)= (10/3)x³−(10/3)x

What are Polynomials?

Before looking at the important questions, it is important to understand what polynomials are:

A polynomial is an algebraic expression that consists of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

General Form: A polynomial in one variable x is expressed as:

P(x)=a_n​xⁿ+a_n-1x^n-1+⋯+a₁x+a₀

Where:

• a_n​,a_n-1​,…,a₁​,a₀​​ are constants (coefficients)
• n is a non-negative integer
• The highest power of the variable in the polynomial is called the degree of the polynomial.

Types of Polynomials

Polynomials can be identified based on the number of terms and the degree:

• Monomial: A polynomial with just one term (e.g.,5x²)
• Binomial: A polynomial with two terms (e.g., 3x²+4x).
• Trinomial: A polynomial with three terms (e.g.,x²+ 5x + 6).

Why Use Class 10 Chapter 2 Important Questions?

Using Class 10 polynomials extra questions has a great impact on practicing and understanding concepts. Here are some of the most important benefits of polynomials extra questions in Class 10:

• Important questions help students focus on the key areas that are more likely to appear in exams. By practicing these questions, students can make sure they cover the most important aspects of the chapter and increase their chances of scoring well.
• Working through important questions allows students to revise their understanding of essential concepts. This practice helps in understanding the relationships between zeroes and coefficients, the division algorithm, and other topics in polynomials.
• Regular practice of important questions builds confidence. When students face similar problems in their exams, they can handle them with ease.
• Practising important questions also benefits time management. Students learn to solve problems more quickly and accurately, which is important when time is limited during the exam.
• By working on important questions, students can identify areas where they are weak or need more practice. 

How to Use Class 10 Chapter 2 Important Questions?

Here are some tips to help you implement these questions in your study schedule:

• Add a set of important questions to your study schedule. Set a dedicated time to work through it each day. This will increase your problem-solving speed. 
• Use these important questions as a self-checking tool. After attempting the questions, check your answers against the solutions. Find any mistakes and understand where you went wrong to avoid repeating them.
• Simulate exam conditions by timing your practice sessions. Attempt to solve the important questions within a set time limit. This will help you get familiar with the pressure of the exam environment.
• Discuss and solve important questions in study groups or with your friends. Group sessions offer new perspectives and different ideas to solve the same problems.
• Regularly solve the important questions, especially those you find challenging. Revision is key to ensuring that the concepts stay fresh in your mind leading up to the exam.
• After practicing important questions, try solving previous years' exam papers. Compare the questions you practiced with those in the actual exam papers. This will help you understand the pattern and difficulty level of the exam.

Scoring good marks in the class 10 mathematics exam will become super easy with the help of class 10 important questions for every chapter. These questions are the best for math, as they require more practice. There are many types of questions at various levels to check your ability to solve difficult questions. Here at Educart, we have provided Class 10 Chapter 2 questions for our students. You can also find the premium class 10 polynomials extra questions in our Educart's one-shot question bank for class 10 math.