Class 10 > Mathematics (Maths) Class 10 > NCERT Solutions: Polynomials (Exercise 2.1)
Polynomials (Exercise 2.1) NCERT Solutions - Mathematics (Maths) Class 10
Q1: The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.
Sol:
Graphical method to find zeroes:
Total number of zeroes in any polynomial equation = total number of times the curve intersects x-axis.
(i) In the given graph, the number of zeroes of p(x) is 0 because the graph is parallel to x-axis does not cut it at any point.
(ii) In the given graph, the number of zeroes of p(x) is 1 because the graph intersects the x-axis at only one point.
(iii) In the given graph, the number of zeroes of p(x) is 3 because the graph intersects the x-axis at any three points.
(iv) In the given graph, the number of zeroes of p(x) is 2 because the graph intersects the x-axis at two points.
(v) In the given graph, the number of zeroes of p(x) is 4 because the graph intersects the x-axis at four points.
(vi) In the given graph, the number of zeroes of p(x) is 3 because the graph intersects the x-axis at three points.
Check out the NCERT Solutions of all the exercises of Polynomials:
Exercise 2.2 NCERT Solutions: Polynomials
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FAQs on Polynomials (Exercise 2.1) NCERT Solutions - Mathematics (Maths) Class 10
| 1. What are the different types of polynomials? |  |
Ans. Polynomials can be classified into various types based on the number of terms they have. Some common types of polynomials include:
- Monomials: These are polynomials with only one term, such as 3x or 5y.
- Binomials: These are polynomials with two terms, such as 2x + 3 or 4y - 7.
- Trinomials: These are polynomials with three terms, such as 2x^2 + 3x + 1 or 4y^2 - 7y + 2.
- Multinomials: These are polynomials with more than three terms, such as 2x^3 + 3x^2 + 4x + 1 or 4y^3 - 7y^2 + 2y + 5.
| 2. What is the degree of a polynomial? | |
Ans. The degree of a polynomial is the highest power of the variable in the polynomial. It helps determine the behavior and characteristics of the polynomial. For example, if a polynomial has the term 3x^2, then the degree of that polynomial is 2. Similarly, if a polynomial has the term 4y^3 + 2y^2 - 5, then the degree of that polynomial is 3.
| 3. How do I add or subtract polynomials? | |
Ans. To add or subtract polynomials, you need to combine like terms. Like terms are terms that have the same variables raised to the same power. Here are the steps to add or subtract polynomials:
1. Arrange the polynomials in vertical columns, aligning the like terms.
2. Add or subtract the coefficients of the like terms.
3. Write the sum or difference of the coefficients along with the common variables and exponents.
For example, to add the polynomials 2x^2 + 3x - 5 and x^2 - 2x + 4, you would arrange them as:
2x^2 + 3x - 5
+ x^2 - 2x + 4
Then, combine the like terms:
2x^2 + x^2 = 3x^2
3x - 2x = x
-5 + 4 = -1
So, the sum of the polynomials is 3x^2 + x - 1.
| 4. How do I multiply polynomials? | |
Ans. To multiply polynomials, you can use the distributive property or the FOIL method. Here are the steps to multiply two polynomials:
1. Multiply each term of one polynomial by each term of the other polynomial.
2. Combine like terms, if any, to simplify the result.
For example, to multiply the polynomials (2x + 3)(x - 4), you would multiply each term as follows:
2x * x = 2x^2
2x * -4 = -8x
3 * x = 3x
3 * -4 = -12
Then, combine the like terms:
2x^2 - 8x + 3x - 12
Simplifying further, we get:
2x^2 - 5x - 12
So, the product of the polynomials is 2x^2 - 5x - 12.
| 5. How can I find the zeros of a polynomial? | |
Ans. To find the zeros of a polynomial, you need to solve the polynomial equation for the variable that makes the polynomial equal to zero. Zeros of a polynomial are also known as roots or solutions of the equation. Here are the steps to find the zeros of a polynomial:
1. Set the polynomial equal to zero.
2. Factorize the polynomial, if possible.
3. Solve the resulting equation by setting each factor equal to zero.
4. Find the values of the variable that satisfy each equation.
For example, to find the zeros of the polynomial x^2 - 5x + 6, we would set it equal to zero:
x^2 - 5x + 6 = 0
Next, we factorize the polynomial:
(x - 2)(x - 3) = 0
Now, we set each factor equal to zero and solve the resulting equations:
x - 2 = 0 or x - 3 = 0
Solving these equations gives us:
x = 2 or x = 3
So, the zeros of the polynomial are x = 2 and x = 3.
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