Class 10 Exam  >  Class 10 Notes  >  RD Sharma Solutions for Class 10 Mathematics  >  Chapter 2 - Polynomials, RD Sharma Solutions - (Part-8)

Chapter 2 - Polynomials, RD Sharma Solutions - (Part-8) | RD Sharma Solutions for Class 10 Mathematics PDF Download

Page No 2.63

Ques.1. Quadratic polynomials, whose zeroes are –4 and 3 are given by ________.
Ans.
Quadratic polynomial with given zeroes is
x2 – (sum of zeroes)x + (product of zeroes)
= x− (−4+3)x +  (−4 × 3)
= x− (−1)x + (−12)
= x+ x − 12
Hence, Quadratic polynomials, whose zeroes are –4 and 3 are given by x2 + x – 12.

Ques.2. Quadratic polynomials with rational coefficients having √3 as a zero are given by ________.
Ans. Irrational zeroes of a quadratic polynomial always occurs in pairs.
If one zero is √3 then, other zero is –√3.
Now,
Quadratic polynomial with given zeroes is
x2 – (sum of zeroes)x + (product of zeroes)
= x2−(√3–+(−√3))x + (√3× − √3)
= x2−(√3 − √3)x + (−3)
= x− 3
Hence, Quadratic polynomials with rational coefficients having √3 as a zero are given by x2 – 3.

Ques.3. The number of quadratic polynomials whose zeroes are 2 and – 3, is __________.
Ans. Quadratic polynomial with given zeroes is  
x– (sum of zeroes)x + (product of zeroes)
=x− (−3 + 2)x + (−3 × 2)
=x− (−1)x + (−6)
=x+ x − 6
Hence, the number of quadratic polynomials whose zeroes are 2 and – 3, is one.

Ques.4. The zeroes of the quadratic polynomial x2 + x – 6 are __________.
Ans.
Let f(x) =  x2 + x – 6
⇒ f(x) = x2 + x − 6
= x2 + 3x − 2x − 6
= x(x + 3) − 2(x + 3)
= (x − 2)(x + 3)
To find the zeroes, we put f(x) = 0
⇒ (x − 2)(x + 3)
⇒ (x − 2) = 0 or (x + 3) = 0
⇒ x = 2, −3
Hence, the zeroes of the quadratic polynomial x2 + x – 6 are 2 and –3.

Ques.5. The quadratic polynomials, the sum and product of whose zeroes are 7 and 12 are given by _________.
Ans. 
Quadratic polynomial with given zeroes is
x2 – (sum of zeroes)x + (product of zeroes)
= x2 − (7)x + (12)
= x2 − 7x + 12
Hence, quadratic polynomials, the sum and product of whose zeroes are 7 and 12 are given by x2 – 7x + 12.

Ques.6. If x + 1 is a factor of the polynomial 2x3 + ax2 + 4x + 1, then a = _________.
Ans. Let f(x) = 2x3 + ax2 + 4x + 1
It is given that one factor of f(x) is (x + 1).
Therefore, f(x) = 0 when x = −1.
On putting x = –1 in f(x) = 0, we get
2(−1)3 + a(−1)2 + 4(−1) + 1 = 0
⇒ 2(−1) + a(+1) + 4(−1) + 1 = 0
⇒ −2 + a − 4 + 1 = 0
⇒ a − 5 = 0
⇒ a = 5
Hence, a = 5.

Ques.7. If a + b + c = 0, then a zero of the polynomial ax2 + bx + c, is _________.
Ans.
Let f(x) = ax2 + bx + c
It is given that a + b + c = 0
Therefore, when x = 1, f(x) = a(1)2 + b(1) + c = a + b + c = 0
Thus, x = 1 is the zero of f(x).
Hence, if a + b + c = 0, then a zero of the polynomial ax2 + bx + c, is 1.

Ques.8. If a + c = b, then a zero of the polynomial ax2 + bx + c, is _________.
Ans. Let f(x) = ax2 + bx + c
It is given that a + c = b
⇒ a − b + c = 0
Therefore, when x = −1, f(x) = a(−1)2 + b(−1) + c = a − b + c = 0
Thus, x = −1 is the zero of f(x).
Hence, if a + c = b, then a zero of the polynomial ax2 + bx + c, is −1.

Ques.9. The graph of a quadratic polynomial intersects the x-axis at the most at ________ points.
Ans.
A quadratic polynomial has at most 2 zeroes.
Thus, it can intersects the x-axis at the most at 2 points.
Hence, the graph of a quadratic polynomial intersects the x-axis at the most at two points.

Ques.10. If two of the zeroes of a cubic polynomial are zero, then it does not have ______ and _______ terms.
Ans.
Let f(x) = ax3 + bx2 + cx + d be a cubic polynomial.
Since, two of the zeroes of a cubic polynomial are zero, then the equation will be ax3 + bx2 = 0
Therefore, it does not have the linear term and the constant term.
Hence, If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.

Ques.11. If all the zeroes of cubic polynomial x3 + ax2 – bx + c are negative then a, b and c all have _______ sign.
Ans.
Let f(x) = x3 + ax2 – bx + c
Let the zeroes of f(x) be α, β, γ,   where all these zeroes are negative
Then,
α + β + γ = −a
Thus, a is positive (∵ α, β, γ are negative)
αβ + βγ + γα = −b
Thus, b is positive (∵ α, β, γ are negative)
αβγ = −c
Thus, c is positive (∵ α, β, γ are negative)
Hence, if all the zeroes of cubic polynomial x3 + ax2 – bx + c are negative then a, b and c all have positive sign.

Ques.12. If the zeroes of the quadratic polynomial ax2 + x + a are equal, then a = ________.
Ans.
Let f(x) = ax2 + x + a
Let the zeroes of f(x) be α and α. (∵the zeroes are equal)
Then,
Chapter 2 - Polynomials, RD Sharma Solutions - (Part-8) | RD Sharma Solutions for Class 10 Mathematics
For α = 1, a = -(1/2)
For α=-1, a= (1/2)
Hence, if the zeroes of the quadratic polynomial ax2 + x + a are equal, then a =Chapter 2 - Polynomials, RD Sharma Solutions - (Part-8) | RD Sharma Solutions for Class 10 Mathematics

Ques.13. If the zeroes of the quadratic polynomial ax2 + bx + c are both negative, then a, b and c all have the _______ sign.
Ans.
Let f(x) = ax2 + bx + c
Let the zeroes of f(x) be α, β, where all these zeroes are negative.
Then,
α+ β = -(b/a)
∵ α, β are negative
∴ b/a is positive
⇒ either both a and b are positive or both are negative.
αβ = c/a
∵ α, β are negative
∴ c/a is positive 
⇒ either both a and c are positive or both are negative.
Hence, if the zeroes of the quadratic polynomial ax2 + bx + c are both negative, then a, b and c all have the same sign.

Ques.14. If x + α is a factor of the polynomial 2x2 + 2αx + 4x + 12, then α = _______.
Ans.
Let f(x) = 2x2 + 2αx + 4x + 12
It is given that one factor of f(x) is (x + α).
Therefore, f(x) = 0 when x = −α.
On putting x = –α in f(x) = 0, we get
2(−α)2 + 2α(−α) + 4(−α) + 12 = 0
⇒2(α)2 − 2α2 − 4α + 12 = 0
⇒ −4α + 12 = 0
⇒ −4α = −12
⇒ α = 3
Hence, if x + α is a factor of the polynomial 2x2 + 2αx + 4x + 12, then α = 3.

Ques.15. If –4 is a zero of the polynomial x2 – x – (2k + 2), then k = _________.
Ans.
Let f(x) = x2 – x – (2k + 2)
It is given that –4 is a zero of f(x).
Therefore, f(x) = 0 when x = −4.
On putting x = –4 in f(x) = 0, we get
(−4)2 − (−4) − (2k + 2) = 0
⇒ 16 + 4 − 2k − 2 = 0
⇒ 20 − 2k − 2 = 0
⇒ 18 − 2k = 0
⇒ 2k = 18
⇒ k = 9
Hence, if –4 is a zero of the polynomial x– x – (2k + 2), then k = 9.

Ques.16. If 4x2 – 6x – m is divisible by x – 3, then m = ________.
Ans.
Let f(x) = 4x2 – 6x – m
It is given that f(x) is divisible by x – 3.
Therefore, f(x) = 0 when x = 3.
On putting x = 3 in f(x) = 0, we get
4(3)2 − 6(3) − m = 0
⇒ 4(9) − 6(3)−m = 0
⇒ 36 − 18 − m = 0
⇒ 18 − m = 0
⇒ m = 18
Hence, if 4x2 – 6x – m is divisible by x – 3, then m = 18.

Ques.17. If a, a + b, a + 2b are zeroes of the cubic polynomial x3 – 6x2 + 3x + 10, then a + b = ________.
Ans. 
Let f(x) = x3 – 6x2 + 3x + 10
Let the zeroes of f(x) be a, a + b, a + 2b.
Then,
Sum of zeroes = 6
⇒ a + (a + b) + (a + 2b) = 6
⇒ 3a + 3b = 6
⇒ 3(a + b) = 6
⇒ a + b = 2
Hence, if a, a + b, a + 2b are zeroes of the cubic polynomial x– 6x+ 3x + 10, then a + b = 2.

Ques.18. If one zero of the quadratic polynomial 2x2 – 6kx + 6x – 7 is negative of the other, then k = ________.
Ans.
Let f(x) = 2x2 – 6kx + 6x – 7 = 2x2 + (– 6k + 6)x – 7
Let the zeroes of f(x) be α and −α. (∵ one zero is negative of the other)
Then,
(α) + (−α) = Chapter 2 - Polynomials, RD Sharma Solutions - (Part-8) | RD Sharma Solutions for Class 10 Mathematics
⇒ α − α = −(−3k + 3)
⇒ 0 = 3k − 3
⇒ 3k = 3
⇒ k = 1
Hence, if one zero of the quadratic polynomial 2x2 – 6kx + 6x – 7 is negative of the other, then k = 1.

Ques.19. If the product of the zeroes of the quadratic polynomial x2 – 3ax + 2a2 – 1 is 7, then a = _______.
Ans.
Let f(x) = x2 – 3ax + 2a2 – 1
Let the zeroes of f(x) be α and β.
Then,
Chapter 2 - Polynomials, RD Sharma Solutions - (Part-8) | RD Sharma Solutions for Class 10 Mathematics

⇒ αβ = 2a2 − 1
⇒ 2a2 − 1 = 7  (∵ product of zeroes is 7)
⇒ 2a2 = 8
⇒ a2 = 4
⇒ a = ±2
Hence, if the product of the zeroes of the quadratic polynomial x2 – 3ax + 2a2 – 1 is 7, then a = ±2.

Ques.20. The sum of the zeros of the quadratic polynomial 2x2 – 3k is ________.
Ans. 
Let f(x) = 2x2 – 3k = 2x2 + 0x – 3k
Let the zeroes of f(x) be α and β.
Then,
α + β = − (0/2)
⇒ α + β = 0
Hence, the sum of the zeroes of the quadratic polynomial 2x2 – 3k is 0.

Ques.21. The parabola representing a quadratic polynomial f(x) = ax2 + bx + c opens upward when _________.
Ans.
Let f(x) = ax2 + bx + c
When a > 0, the parabola representing f(x) opens upward.
When a < 0, the parabola representing f(x) opens downward.
Hence, the parabola representing a quadratic polynomial f(x) = ax2 + bx + c opens upward when a is positive.

Ques.22. The parabola representing a quadratic polynomial f(x) = ax2 + bx + c opens downward when __________.
Ans.
Let f(x) = ax2 + bx + c
When a > 0, the parabola representing f(x) opens upward.
When a < 0, the parabola representing f(x) opens downward.
Hence, the parabola representing a quadratic polynomial f(x) = ax2 + bx + c opens downward when a is negative.

Ques.23. If the parabola represented by f(x) = ax2 + bx + c cuts x-axis at two distinct points, then the polynomial ax2 + bx + c has ________ real zeroes.
Ans. 
Let f(x) = ax2 + bx + c
If f(x) has two real and distinct zeroes, the parabola represented by f(x) cuts x-axis at two distinct points.
If f(x) has two real and equal zeroes, the parabola represented by f(x) touches x-axis at only one distinct point.
Hence, if the parabola represented by f(x) = ax2 + bx + c cuts x-axis at two distinct points, then the polynomial ax2 + bx + c has 2 real zeroes.

Ques.24. The maximum number of zeroes which a quadratic polynomial can have is _________.
Ans.
Let f(x) = ax2 + bx + c
Maximum number of zeroes of polynomial =  Highest power of x = 2
Therefore, It has at most 2 zeroes.
Hence, the maximum number of zeroes which a quadratic polynomial can have is 2.


Page No 2.64

Ques.1. Define a polynomial with real coefficients.
Ans.
In the polynomial f(x) = anxn + an - 1xn - 1 + .... + a1x + a0,
anxn, an-1xn-1, ...,a1x, and aare known as the terms of the polynomial and an, an-1,...,aand a0 are their real coefficients.
For example, p(x) = 3x - 2 is a polynomial and 3 is a real coefficient

Ques.2. Define degree of a polynomial.
Ans.
The exponent of the highest degree term in a polynomial is known as its degree.
In other words, the highest power of x in a polynomial f(x) is called the degree of the polynomial f(x).
For Example: g(x) = 2x2 + 3x + 4 is a polynomial in the variable x of degree 2.

Ques.3. Write the standard form of a linear polynomial with real coefficients.
Ans.
Any linear polynomial in variable x with real coefficients is of the form f(x) = ax + b, where a, b are real numbers and a ≠ 0.

Ques.4. Write the standard form of a quadratic polynomial with real coefficients.
Ans.
Any quadratic polynomial in variable x with real coefficients is of the form f(x) = ax2 + bx + c, where a, b, c are real numbers and a ≠ 0.

Ques.5. Write the standard form of a cubic polynomial with real coefficients.
Ans. The most general form of a cubic polynomial with coefficients as real numbers is of the form f(x) = ax3 + bx2 + cx + d, where a, b, c, d are real number and a ≠ 0.

The document Chapter 2 - Polynomials, RD Sharma Solutions - (Part-8) | RD Sharma Solutions for Class 10 Mathematics is a part of the Class 10 Course RD Sharma Solutions for Class 10 Mathematics.
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FAQs on Chapter 2 - Polynomials, RD Sharma Solutions - (Part-8) - RD Sharma Solutions for Class 10 Mathematics

1. What are the different types of polynomials?
Ans. Polynomials can be classified into different types based on the number of terms they have. Some common types of polynomials include monomials (one term), binomials (two terms), trinomials (three terms), and polynomials with more than three terms.
2. How can we determine the degree of a polynomial?
Ans. The degree of a polynomial is the highest power of the variable in the polynomial. To determine the degree, identify the term with the highest power of the variable and that will be the degree of the polynomial.
3. What is the difference between a constant and a variable in a polynomial?
Ans. In a polynomial, a constant is a term that does not contain any variable, while a variable is a term that contains a letter representing a number. Constants have a fixed value, while variables can take on different values.
4. Can a polynomial have negative exponents?
Ans. No, a polynomial cannot have negative exponents. A polynomial is an algebraic expression consisting of variables, constants, and exponents that are non-negative integers. Negative exponents would make the expression a non-polynomial.
5. How can we add or subtract polynomials?
Ans. To add or subtract polynomials, align like terms and combine their coefficients. Add or subtract the coefficients of the like terms while keeping the variables and exponents the same.
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