Important Questions Class 10 Maths Chapter 2 Polynomials
In this article explore Important Questions Class 10 Maths Chapter 2 Polynomials. Polynomial is a mathematical expression that consists of variables and coefficients involved in mathematical calculations like addition, subtraction, and multiplication, especially the addition and positive integer powers of variables. Apart from holding importance in mathematics they are also widely used in Chemistry, Physics, Economics, Social science studies, and other critical scientific studies. Graphical representation or algebraic geometry of polynomials are interesting fields to explore.
The numeric zero ‘0’ and its relationship with coefficients hold great importance while studying polynomials. Students are required to pay attention while understanding various degrees (eg. linear, quadratic, cubic) of polynomials.
Important questions for class 10 maths Chapter 2 Polynomials might be circled around the undermentioned topics:
- Quadratic polynomial
- Cubic polynomial
- Relationship between zeroes and coefficients of polynomials
- Graphic representation of polynomials
Important Questions Chapter 2 – Polynomials
Q1. If α and β are the zeroes of a polynomial such that α + β = -6 and αβ = 5, then find the polynomial.
Solution:
Quadratic polynomial is x^2 – Sx + P = 0
⇒ x^2 – (-6)x + 5 = 0
⇒ x^2 + 6x + 5 = 0
Q2. Form a quadratic polynomial whose zeroes are 3 + √2 and 3 – √2.
Solution:
Sum of zeroes,
S = (3 + √2) + (3 – √2) = 6
Product of zeroes,
P = (3 + √2) x (3 – √2) = (3)^2 – (√2)^2 = 9 – 2 = 7
Quadratic polynomial = x^2 – Sx + P = x^2 – 6x + 7
Q3. Find a quadratic polynomial, the sum and product of whose zeroes are 0 and -√2 respectively.
Solution:
Quadratic polynomial is
x^2 – (Sum of zeroes) x + (Product of zeroes)
= x^2 – (0)x + (-√2)
= x^2 – √2
Q4. How many zeros does the polynomial (x – 3)^2 – 4 have? Also, find its zeroes.
Solution:
Given polynomial is (x – 3)^2 – 4
Now, expand this expression.
=> x^2 + 9 – 6x – 4
= x^2 – 6x + 5
As the polynomial has a degree of 2, the number of zeroes will be 2.
Now, solve x^2 – 6x + 5 = 0 to get the roots.
So, x^2 – x – 5x + 5 = 0
=> x(x – 1) -5(x – 1) = 0
=> (x – 1)(x – 5) = 0
x = 1, x = 5
So, the roots are 1 and 5.
Q5. Find the quadratic polynomial if its zeroes are 0, √5.
Solution:
A quadratic polynomial can be written using the sum and product of its zeroes as:
x^2 – (α + β)x + αβ
Where α and β are the roots of the polynomial.
Here, α = 0 and β = √5
So, the polynomial will be:
x^2 – (0 + √5)x + 0(√5)
= x^2 – √5x
Q6. Find the value of “x” in the polynomial 2a^2 + 2xa + 5a + 10 if (a + x) is one of its factors.
Solution:
Let f(a) = 2a^2 + 2xa + 5a + 10
Since, (a + x) is a factor of 2a^2 + 2xa + 5a + 10, f(-x) = 0
So, f(-x) = 2x^2 – 2x^2 – 5x + 10 = 0
-5x + 10 = 0
5x = 10
x = 10/5
Therefore, x = 2
Q7. Can (x – 2) be the remainder on division of a polynomial p(x) by (2x + 3)? Justify your answer.
Solution:
In case of division of a polynomial by another polynomial, the degree of the remainder (polynomial) is always less than that of the divisor. (x – 2) can not be the remainder when p(x) is divided by (2x + 3) as the degree is the same.
Q8. Find a quadratic polynomial, the sum and product of whose zeroes are -8 and 12 respectively. Hence find the zeroes.
Solution:
Let Sum of zeroes (α + β) = S = -8 …[Given]
Product of zeroes (αβ) = P = 12 …[Given]
Quadratic polynomial is x^2 – Sx + P
= x^2 – (-8)x + 12
= x^2 + 8x + 12
= x^2 + 6x + 2x + 12
= x(x + 6) + 2(x + 6)
= (x + 2)(x + 6)
Zeroes are:
x + 2 = 0 or x + 6 = 0
x = -2 or x = -6
Q9. If one zero of the quadratic polynomial f(x)=4x^2−8kx−9 is negative of the other, find the value of ‘k’.
Solution:
It is given that one zero of the quadratic polynomial f(x)=4x^2−8kx−9 is the negative of the other,
Let us take one zero to be α, then the other is −α.
Hence, the sum of the zeros =0
⇒8k/4=0
⇒k=0
Q10. If one zero of the polynomial 5z2 + 13z – p is reciprocal of the other, then find p.
Solution : -5
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