All questions of Polynomials for Class 10 Exam

The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the polynomial is 4x^7 + 9x^5 + 5x^2 + 11.

To determine the degree of this polynomial, we need to look for the term with the highest power of x.

- Identify the terms and their degrees
The polynomial consists of four terms: 4x^7, 9x^5, 5x^2, and 11. The degrees of these terms are as follows:
- 4x^7 has a degree of 7
- 9x^5 has a degree of 5
- 5x^2 has a degree of 2
- 11 is a constant term and has a degree of 0.

- Compare the degrees and determine the highest degree
Since the degree of 7 (from the term 4x^7) is the highest among all the terms in the polynomial, the degree of the polynomial is 7.

- Explain the correct answer
Therefore, the correct answer is option 'A', which states that the degree of the polynomial is 7.

√x + 3x - 7

B

Explanation:

For a polynomial of degree one, the general form is given by p(x) = ax + b, where 'a' and 'b' are constants.

To understand why option 'A' is the correct answer, let's consider the given condition p(y) = 0.

Zero of p(x):
A zero (or root) of a polynomial is a value of the variable that makes the polynomial equal to zero. In other words, if p(x) = 0, then 'x' is a zero of p(x).

In this case, p(y) = 0, which means that the polynomial is equal to zero when x = y. So, 'y' is a zero of p(x).

Therefore, option 'A' is the correct answer.

Other options:

Value of p(x):
The value of a polynomial at a particular point is obtained by substituting that point into the polynomial. In this case, p(y) represents the value of the polynomial at the point x = y. However, we are not given any information about the value of p(x) or y, so option 'B' is incorrect.

Constant of p(x):
The constant term of a polynomial is the term that does not have any variable associated with it. In the given polynomial p(x) = ax + b, the constant term is 'b'. However, we are not given any information about the constant term or its relation to y, so option 'C' is incorrect.

Conclusion:
Based on the given condition p(y) = 0, the correct answer is option 'A' - y is a zero of p(x).

Understanding the Problem
To find the quadratic polynomial whose zeros are 3 + √2 and 3 - √2, we can use the relationship between the zeros and coefficients of a quadratic polynomial.
Forming the Polynomial
A quadratic polynomial can be expressed in the standard form as:
P(x) = x² - (sum of the zeros)x + (product of the zeros)
Calculating the Sum of the Zeros
- The zeros are 3 + √2 and 3 - √2.
- Sum = (3 + √2) + (3 - √2) = 6.
Calculating the Product of the Zeros
- Product = (3 + √2)(3 - √2)
- Using the difference of squares:
= 3² - (√2)² = 9 - 2 = 7.
Constructing the Polynomial
Now, substitute the sum and product back into the polynomial formula:
P(x) = x² - (sum of zeros)x + (product of zeros)
P(x) = x² - 6x + 7.
Conclusion
The quadratic polynomial with the given zeros is:
P(x) = x² - 6x + 7.
Thus, the correct answer is option 'C'.

The discriminant of a quadratic polynomial is a key factor in determining the nature of its roots. The discriminant, denoted as D, is calculated as the expression b² - 4ac, where a, b, and c are the coefficients of the quadratic polynomial in the standard form ax² + bx + c.

When the discriminant is greater than zero (D > 0), it implies that the quadratic polynomial has two real and unequal roots. This can be explained in detail as follows:

1. Definition of Discriminant:
The discriminant is a mathematical term used to determine the nature of the roots of a quadratic polynomial. It is calculated using the formula D = b² - 4ac, where a, b, and c are the coefficients of the quadratic polynomial.

2. Nature of Discriminant:
The discriminant can take three different values:

- If the discriminant is greater than zero (D > 0), it means that the quadratic polynomial has two real and unequal roots.
- If the discriminant is equal to zero (D = 0), it means that the quadratic polynomial has two real and equal roots.
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3. Explanation of Option B:
In the given question, the discriminant is stated to be greater than zero (D > 0). According to the nature of the discriminant, this implies that the quadratic polynomial has two real and unequal roots. Therefore, the correct answer is option B.

4. Example:
Consider the quadratic polynomial x² - 5x + 6. To determine the nature of its roots, we can calculate the discriminant using the formula D = b² - 4ac. Here, a = 1, b = -5, and c = 6.

D = (-5)² - 4(1)(6) = 25 - 24 = 1

Since the discriminant (D = 1) is greater than zero, the quadratic polynomial has two real and unequal roots.

By considering the nature of the discriminant, we can conclude that when the discriminant is greater than zero (D > 0), the quadratic polynomial has two real and unequal roots.

Introduction:
In this question, we are given a polynomial and asked to determine its degree based on the number of points where it intersects the x-axis. We need to explain why the correct answer is option 'C' - cubic polynomial.

Explanation:
A polynomial is an algebraic expression consisting of variables and coefficients. The degree of a polynomial is determined by the highest power of the variable in the expression. Let's consider each option and analyze why only a cubic polynomial can intersect the x-axis at 3 points.

a) Linear Polynomial:
A linear polynomial has a degree of 1 and is in the form of "ax + b". It represents a straight line on the graph. A linear polynomial can intersect the x-axis at most once. Therefore, it cannot satisfy the condition of intersecting the x-axis at 3 points.

b) Quadratic Polynomial:
A quadratic polynomial has a degree of 2 and is in the form of "ax^2 + bx + c". It represents a parabola on the graph. A quadratic polynomial can intersect the x-axis at most twice. Therefore, it also cannot satisfy the condition of intersecting the x-axis at 3 points.

c) Cubic Polynomial:
A cubic polynomial has a degree of 3 and is in the form of "ax^3 + bx^2 + cx + d". It represents a curve on the graph. A cubic polynomial can intersect the x-axis at most three times. Therefore, it satisfies the condition of intersecting the x-axis at 3 points.

d) Biquadratic Polynomial:
A biquadratic polynomial has a degree of 4 and is in the form of "ax^4 + bx^3 + cx^2 + dx + e". It represents a curve on the graph. A biquadratic polynomial can intersect the x-axis at most four times. Therefore, it exceeds the condition of intersecting the x-axis at 3 points.

Conclusion:
Based on the analysis above, it can be concluded that if the graph of a polynomial intersects the x-axis at 3 points, then the polynomial must be a cubic polynomial. Hence, the correct answer is option 'C'.

Explanation:

The correct answer is option 'B' - "Intersects x-axis".

To understand why this is the correct answer, let's first define what a polynomial zero is. In algebra, a zero of a polynomial function is a value of the variable that makes the polynomial equal to zero. In other words, it is a value of 'x' that satisfies the equation f(x) = 0.

Now, let's consider the graph of a polynomial function. The graph of a polynomial is a smooth curve that can intersect the x-axis at various points. Each point of intersection represents a value of 'x' for which the polynomial is equal to zero, i.e., a zero of the polynomial.

Graphical Representation:

When we graph a polynomial function, the x-axis represents the values of 'x', and the y-axis represents the corresponding values of the polynomial function, f(x).

Number of Polynomial Zeros:

The number of polynomial zeros is equal to the number of points where the graph of the polynomial intersects the x-axis. This is because these points represent the values of 'x' for which the polynomial is equal to zero.

Intersects Y-Axis:

The y-axis is the vertical line on the graph where x = 0. It is essential to note that the y-axis does not represent the values of 'x'. Therefore, the points where the graph intersects the y-axis do not correspond to the zeros of the polynomial. Instead, they represent the constant term or the y-intercept of the polynomial function.

Intersects X-Axis:

The x-axis is the horizontal line on the graph where y = 0. The points where the graph intersects the x-axis represent the values of 'x' for which the polynomial is equal to zero. Therefore, these points correspond to the zeros of the polynomial function.

Intersects Y-Axis or X-Axis:

The option 'C' incorrectly suggests that the number of polynomial zeros is equal to the number of points where the graph intersects either the y-axis or the x-axis. As explained earlier, the points of intersection with the y-axis do not correspond to the zeros of the polynomial. Therefore, this option is incorrect.

None of the Above:

The option 'D' is incorrect as well because, as mentioned earlier, the number of polynomial zeros is equal to the number of points where the graph intersects the x-axis. So, there are points on the graph that represent the zeros of the polynomial.

In conclusion, the correct answer is option 'B' - "Intersects x-axis" because the number of polynomial zeros is equal to the number of points where the graph of the polynomial intersects the x-axis.

Definition of Zeros of a Polynomial:
Zeros of a polynomial are the real numbers that make the polynomial equal to zero when substituted for the variable.

Explanation of the Correct Answer:

Option D: p(α) = 0
This option is the correct answer because the zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, if a real number α is a zero of the polynomial p(x), then p(α) = 0. This is a fundamental property of zeros of polynomials.
Therefore, the correct statement to define zeros of a polynomial is p(α) = 0.

Understanding the Other Options:
- Option A: p(α) = 4
This statement does not define the zeros of the polynomial. Zeros of a polynomial are the values that make the polynomial equal to zero, not any other constant.
- Option B: p(α) = 1
Similarly, this statement does not define the zeros of the polynomial. Zeros are the values that make the polynomial equal to zero, not 1.
- Option C: p(α) ≠ 0
This statement does not accurately define the zeros of the polynomial. Zeros are the values that make the polynomial equal to zero, not any value other than zero.

In Conclusion:
The correct definition of zeros of a polynomial is when the polynomial evaluated at a real number α results in p(α) = 0. This is a fundamental concept in algebra and is crucial in understanding the roots of polynomial functions.

Polynomial of Degree p:
- A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.
- The degree of a polynomial is the highest power of the variable in the polynomial.
- A polynomial of degree p can be written as:
P(x) = a_px^p + a_{p-1}x^{p-1} + ... + a_1x + a_0

Number of Zeroes:
- Zeroes of a polynomial are the values of x for which the polynomial evaluates to zero.
- In other words, a value x is a zero of a polynomial if P(x) = 0.
- The number of zeroes of a polynomial can be equal to or less than the degree of the polynomial.

Option D - At most p zeroes:
- This option means that the polynomial can have a maximum of p zeroes.
- It indicates that the number of zeroes of the polynomial is not necessarily equal to the degree of the polynomial.
- Let's consider an example to understand this concept.

Example:
- Consider a polynomial of degree 3: P(x) = x^3 - 2x^2 + x - 1.
- We can find the zeroes of this polynomial by setting P(x) = 0 and solving for x.
- However, in this case, the polynomial may not have exactly 3 zeroes.
- It could have fewer zeroes depending on the nature of the polynomial.
- In this example, the polynomial has only 1 zero: x = 1.
- Therefore, this example illustrates that a polynomial of degree 3 can have at most 3 zeroes, but it may have fewer zeroes.

Conclusion:
- A polynomial of degree p can have at most p zeroes.
- The number of zeroes can be less than p, depending on the nature of the polynomial.
- Option D, "At most p zeroes," correctly describes the possible number of zeroes for a polynomial of degree p.