Worksheet Solutions: Polynomials

# Polynomials Class 10 Worksheet Maths Chapter 2

Q1: If a and b are roots of  the equation x2 + 7 x + 7 . Find the value of a−1 + b−1 − 2αb ?
Ans: for f ( x ) = x2 + 7 x + 7
we get
α + b = − 7
αb = 7
Now a−1 + b−1 − 2αb

Q2: If the zeroes of the quadratic polynomial x2 + (α + 1 ) x + b are 2 and -3, then find the value of a and b.
Ans: Let f (x) = x2 + (a + 1) x + b
Then 2 − 3 = − ( a + 1 ) or α = 0
− 6 = b
So a = 0 and b = 6

Q.3. If a and b are zeroes of the polynomial f (x) = 2x2 − 7x + 3, find the value of α2 + b2.

Ans: f (x) = 2x2 − 7 x + 3
= 2x2 − x − 6 x + 3 = x(2x − 1) − 3(2 x − 1) = (x − 3) (2 x − 1)
So zeroes are 3 and 1/2
Now
α2 + b2

Q.4. Find the zeroes of the quadratic polynomial x2 + x − 12 and verify the relationship between the zeroes and the coefficients.
Ans: x2 + x − 12
= x2 + 4x − 3x − 12
= x (x + 4) − 3 (x + 4) = (x − 3) (x + 4)
So zeroes are 3 and -4
as we know
sum of roots = -b/a = -(1)/1 = -1 ie 3+(-4) = -1
product of roots = c/a = -12/1 = -12 ie 3x-4 = -12

Q5: If p and q are zeroes of f (x) = x2 − 5x + k, such that p − q = 1 , find the value of k.
Ans: for f ( x ) = x2 − 5 x + k
we get p + q = 5
pq = k
Now p − q = 1
(p − q)2 = 1
(p + q)2 − 4pq = 1
25 − 4k = 1
k = 6

Q6: Given that two of the zeroes of the cubic polynomial αx3 + bx2 + cx + d are 0, then find the third zero.
Ans: Two zeroes = 0, 0
Let the third zero be k.
The, using relation between zeroes and coefficient of polynomial, we have:
k + 0 + 0 = − b a
Third zero = k = -b/a

Q.7. If one of the zeroes of the cubic polynomial x3 + αx2 + bx + c is -1, then find the product of the other two zeroes.
Ans:

Q8: If a-b, a a+b , are zeroes of x3 − 6x2 + 8x , then find the value of b
Ans:
Let f ( x ) = x3 − 6x2 + 8 x
Method -1
= x (x2 − 6 x + 8) = x(x − 2) (x − 4)
So 0,2,4 are zeroes of the polynomial. or a=2 and b=2 or -2 Method -2

Q9: Quadratic polynomial 4x2 + 12x + 9 has zeroes as p and q . Now form a quadratic polynomial whose zeroes are p − 1 and q − 1
Ans:
4x2 + 12 x + 9
= 4x2 + 6x + 6x + 9
= 2x(2x + 3) + 3(2x + 3) = (2 x + 3)2
So p = -3/2 and q = -3/2
So, p − 1 = − 5/2 and q − 1 = − 5/2
(x + 5/2)2
or
4x2 + 20 x + 25

Q10: p and q are zeroes of the quadratic polynomial  x2 − (k + 6 ) x + 2(2 k − 1) . Find the value of k if 2(p + q) = p q
Ans: for f (x) = x2 − (k + 6) x + 2 (2 k − 1)
We get, p + q = k + 6
p q = 2(2 k − 1)
Now 2 (p + q ) = pq
Therefore,
2 (k + 6) = 2(2k − 1)
or k + 6 = 2 k − 1
or k = 7

Q11: Given that the zeroes of the cubic polynomial x3 − 6 x2 + 3 x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.
Ans: k1 + k2 + k3 =
a + a + b + a + 2b = 6
a + b = 2
a = 2 − b
Now

b=-3 or b=3
So a= 5 or -1
The zeroes with a = 5, b= -3 can be expressed as 5, 2, -1
The zeroes with a = -1, b = 3 can be expressed as -1, 2, 5

Q12: If one zero of the polynomial 2x2−5x−(2k + 1) is twice the other, find both the zeroes of the polynomial and the value of k.

Ans: Let a be one zero ,then another will be 2a

Now
α + 2α = 5/2 or a= 5/6
Also

k= -17/9

Q.13: Using division show that 3y2 + 5 is a factor of 6y5 + 15y4 + 16y3 + 4y2 + 10y − 35 .
Ans:

Q14: If (x - 2) and [x - 1/2 ] are the factors of the polynomials qx2 + 5x + r prove that q = r.
Ans:
4q + 10 + r = 0 -(1)
q/4 +5/2 + r = 0 or q + 10 + 4r = 0 -(2)
Subtracting 1 from 2
3q-3r = 0
q = r

Q15: Find k so that the polynomial x2 + 2x + k is a factor of polynomial 2x4 + x3 - 14x2 + 5x + 6. Also, find all the zeroes of the two polynomials.
Ans:
For x2 + 2x + k is a factor of polynomial 2x4 + x3 - 14x2 + 5x + 6, it should be able to divide the polynomial without any remainder

Comparing the coefficient of x we get.
21 + 7k = 0
k = -3
So x2 + 2x + k becomes x2 + 2x -3 = (x-1)(x+3)
Now
2x4 + x3 - 14x2 + 5x + 6= (x2 + 2x -3)(2x2-3x-8+2k)
=(x2 + 2x - 3)(2x2-3x - 2)
=(x - 1)(x + 3)(x - 2)(2x + 1)
or x = 1, -3, 2,= -1/2

The document Polynomials Class 10 Worksheet Maths Chapter 2 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## Mathematics (Maths) Class 10

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## FAQs on Polynomials Class 10 Worksheet Maths Chapter 2

 1. What are polynomials and how are they classified?
Ans. Polynomials are mathematical expressions that consist of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. They are classified based on their degree (the highest power of the variable) and the number of terms. For example, a polynomial with one term is called a monomial, two terms a binomial, and three or more terms a polynomial.
 2. How do you add and subtract polynomials?
Ans. To add or subtract polynomials, you combine like terms, which are terms that have the same variable raised to the same power. For addition, you simply add the coefficients of the like terms together, and for subtraction, you subtract the coefficients. The result will be a new polynomial.
 3. What is the process for multiplying polynomials?
Ans. To multiply polynomials, you use the distributive property, often referred to as the FOIL method for binomials. You multiply each term in the first polynomial by each term in the second polynomial and then combine like terms to simplify the result.
 4. How can polynomials be factored?
Ans. Polynomials can be factored by finding common factors, using techniques such as grouping, or applying special formulas like the difference of squares or the quadratic formula for quadratic polynomials. The goal is to express the polynomial as a product of simpler polynomials.
 5. What are the applications of polynomials in real life?
Ans. Polynomials have various applications in real life, including modeling physical phenomena such as projectile motion, calculating areas and volumes in geometry, and in economics for optimization problems. They are also used in computer graphics and data fitting techniques in statistics.

## Mathematics (Maths) Class 10

116 videos|420 docs|77 tests

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