# Class 8 Maths Chapter 2 Question Answers - Polynomials

Q1: If the polynomial (x4 + 2x3 + 8x2 + 12x + 18) is divided by another polynomial (x2 + 5), the remainder comes out to be (px + q), find the values of p and q.
Ans:

Remainder = 2x + 3
px + q = 2x + 3
p = 2 and q = 3.

Q2: If a polynomial 3x– 4x3 – 16x2 + 15x + 14 is divided by another polynomial x2 – 4, the remainder comes out to be px + q. Find the value of p and q.
Ans:

Q3: Find the values of a and b so that x4 + x3 + 8x2 +ax – b is divisible by x2 + 1.
Ans:

If x4 + x3 + 8x2 + ax – b is divisible by x+ 1
Remainder = 0
(a – 1)x – b – 7 = 0
(a – 1)x + (-b – 7) = 0 . x + 0
a – 1 = 0, -b – 7 = 0
a = 1, b = -7
a = 1, b = -7

Q4: What must be subtracted from p(x) = 8x4 + 14x3 – 2x2 + 8x – 12 so that 4x2 + 3x – 2 is factor of p(x)? This question was given to group of students for working together.
Ans:

Polynomial to be subtracted by (15x – 14).

Q5: If α and β are the zeroes of the polynomial p(x) = 2x+ 5x + k, satisfying the relation, α2 + β2 + αβ = 21/4 then find the value of k.
Ans:
Given polynomial is p(x) = 2x2 + 5x + k
Here a = 2, b = 5, c = k

Q6: If α and β are zeroes of p(x) = kx2 + 4x + 4, such that α2 + β2 = 24, find k.
Ans:
We have, p(x) = kx2 + 4x + 4
Here a = k, b = 4, c = 4

⇒ 24k2 = 16 – 8k
⇒ 24k2 + 8k – 16 = 0
⇒ 3k2 + k – 2 = 0 …[Dividing both sides by 8]
⇒ 3k2 + 3k – 2k – 2 = 0
⇒ 3k(k + 1) – 2(k + 1) = 0
⇒ (k + 1)(3k – 2) = 0
⇒ k + 1 = 0 or 3k – 2 = 0
⇒ k = -1 or k = 2/3

Q7: If p(x) = x3 – 2x2 + kx + 5 is divided by (x – 2), the remainder is 11. Find k. Hence find all the zeroes of x3 + kx2 + 3x + 1.
Ans:

p(x) = x3 – 2x2 + kx + 5,
When x – 2,
p(2) = (2)3 – 2(2)2 + k(2) + 5
⇒ 11 = 8 – 8 + 2k + 5
⇒ 11 – 5 = 2k
⇒ 6 = 2k
⇒ k = 3
Let q(x) = x3 + kx2 + 3x + 1
= x3 + 3x2 + 3x + 1
= x3 + 1 + 3x2 + 3x
= (x)3 + (1)3 + 3x(x + 1)
= (x + 1)3
= (x + 1) (x + 1) (x + 1) …[∵ a3 + b3 + 3ab (a + b) = (a + b)3]
All zeroes are:
x + 1 = 0 ⇒ x = -1
x + 1 = 0 ⇒ x = -1
x + 1 = 0 ⇒ x = -1
Hence zeroes are -1, -1 and -1.

Q8: Find all the zeroes of the polynomial 8x4 + 8x3 – 18x2 – 20x – 5, if it is given that two of its zeroes are  and
Ans:

Q9: If a polynomial x4 + 5x3 + 4x2 – 10x – 12 has two zeroes as -2 and -3, then find the other zeroes.
Ans:

Since two zeroes are -2 and -3.
(x + 2)(x + 3) = x2 + 3x + 2x + 6 = x2 + 5x + 6
Dividing the given equation with x2 + 5x + 6, we get

x4 + 5x+ 4x2 – 10x – 12
= (x2 + 5x + 6)(x2 – 2)
= (x + 2)(x + 3)(x – √2 )(x + √2 )
Other zeroes are:
x – √2 = 0 or x + √2 = 0
x = √2 or x = -√2

Q10: Given that x – √5 is a factor of the polynomial x3 – 3√5 x2 – 5x + 15√5, find all the zeroes of the polynomial.
Ans:

Let P(x) = x3 – 3√5 x2 – 5x + 15√5
x – √5 is a factor of the given polynomial.
Put x = -√5,

Other zero:
x – 3√5 = 0 ⇒ x = 3√5
All the zeroes of P(x) are -√5, √5 and 3√5.

The document Class 8 Maths Chapter 2 Question Answers - Polynomials is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## FAQs on Class 8 Maths Chapter 2 Question Answers - Polynomials

 1. What are polynomials and how are they classified?
Ans. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They are classified based on the number of terms they contain - monomials (1 term), binomials (2 terms), trinomials (3 terms), and so on.
 2. How do you add and subtract polynomials?
Ans. To add or subtract polynomials, simply combine like terms. Like terms have the same variables raised to the same exponents. Add or subtract the coefficients of these like terms while keeping the variables the same.
 3. What is the degree of a polynomial and how is it determined?
Ans. The degree of a polynomial is determined by the highest exponent of the variables in the polynomial. For example, in the polynomial 3x^2 + 5x + 1, the highest exponent is 2, so the degree of this polynomial is 2.
 4. How can you multiply polynomials using the distributive property?
Ans. To multiply polynomials, use the distributive property, where each term of one polynomial is multiplied by each term of the other polynomial. Then, combine like terms. This process is often simplified using the FOIL method for binomials.
 5. Can polynomials have negative exponents?
Ans. No, polynomials cannot have negative exponents. Negative exponents indicate the presence of terms with negative powers, which do not fall under the definition of polynomials. Polynomials only include terms with non-negative integer exponents.

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