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1: GST [Goods and Service Tax]
2: Banking (Recurring Deposit Account)
3: Shares and Dividend
Unit 2. Algebra
4: Linear Inequations (In one variable)
5: Quadratic Equations
6: Solving (simple) Problems (Based on Quadratic Equations)
7: Ratio and Proportion (Including Properties and Uses)
▶ 8: Factorization of Polynomials (Remainder and Factor Theorems)
9: Matrices
10: Arithmetic Progression
11: Geometric Progression
Unit 3. Co-ordinate Geometry
12: Reflection
13: Section Formula and Mid-Point Formula
14: Equation of a Line
Unit 4. Geometry
15: Similarity (With Applications to Maps and Models)
16: Loci (Locus and Its Constructions)
17: Circles
18: Tangents and Intersecting Chords
19: Constructions (Circles)
Unit 5. Mensuration
20: Cylinder, Cone and Sphere
Unit 6. Trigonometry
21: Trigonometrical Identities
22: Height and Distances
Unit 7. Statistics
23: Graphical Representation
24: Measure of Central Tendency (Mean, Median, Quartiles and Mode)
25: Probability
![Selina solutions for Concise Mathematics [English] Class 10 ICSE chapter 8 - Factorization of Polynomials (Remainder and Factor Theorems) - Shaalaa.com](/images/concise-mathematics-english-class-10-icse_6:7eb8c97e7ccc4a1c956f7ac8305d25e3.jpg "Selina solutions for Concise Mathematics [English] Class 10 ICSE chapter 8 - Factorization of Polynomials (Remainder and Factor Theorems)")
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Solutions for Chapter 8: Factorization of Polynomials (Remainder and Factor Theorems)
Below listed, you can find solutions for Chapter 8 of CISCE Selina for Concise Mathematics [English] Class 10 ICSE.
Selina solutions for Concise Mathematics [English] Class 10 ICSE 8 Factorization of Polynomials (Remainder and Factor Theorems) Exercise 8 (A) [Page 104]
Multiple Choice Type: Choose the correct answer from the options given below.
x – 1 is a factor of 8x2 – 7x + m; the value of m is ______.
–1
1
–2
2
If (x − a) is a factor off(x) then the remainder when f(x) - k is divided by (x − a) is ______.
0
k
−k
a
One factor of x3 – kx2 + 11x – 6 is x – 1. The value of k is ______.
– 6
12
6
–12
If (x – a) is a factor of x3 – ax2 + x + 5; the value of a is ______.
`1/5`
5
`-1/5`
–5
(x – 2) is a factor of ______.
x3 – x2 + x – 6
x3 + x2 + x + 6
2x3 – 6x2 + 5x – 1
x3 – 4x2 + x – 8
Find the remainder when x4 – 3x2 + 2x + 1 is divided by x – 1.
Find the remainder when x3 + 3x2 – 12x + 4 is divided by x – 2.
Find the remainder when x4 + 1 is divided by x + 1.
Use the Remainder Theorem to find which of the following is a factor of 2x3 + 3x2 – 5x – 6.
x + 1
Use the Remainder Theorem to find which of the following is a factor of 2x3 + 3x2 – 5x – 6.
2x – 1
If 2x + 1 is a factor of 2x2 + ax – 3, find the value of a.
Find the values of constants a and b when x – 2 and x + 3 both are the factors of expression x3 + ax2 + bx – 12.
Find the value of k, if 2x + 1 is a factor of (3k + 2)x3 + (k − 1).
Find the values of m and n so that x – 1 and x + 2 both are factors of x3 + (3m + 1)x2 + nx – 18.
When x3 + 2x2 – kx + 4 is divided by x – 2, the remainder is k. Find the value of constant k.
Find the value of a, if the division of ax3 + 9x2 + 4x – 10 by x + 3 leaves a remainder 5.
If x3 + ax2 + bx + 6 has x – 2 as a factor and leaves a remainder 3 when divided by x – 3, find the values of a and b.
What number should be added to 3x3 – 5x2 + 6x so that when resulting polynomial is divided by x – 3, the remainder is 8?
What number should be subtracted from x3 + 3x2 – 8x + 14 so that on dividing it by x – 2, the remainder is 10?
The polynomials 2x3 – 7x2 + ax – 6 and x3 – 8x2 + (2a + 1)x – 16 leaves the same remainder when divided by x – 2. Find the value of ‘a’.
If (x – 2) is a factor of the expression 2x3 + ax2 + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.
Selina solutions for Concise Mathematics [English] Class 10 ICSE 8 Factorization of Polynomials (Remainder and Factor Theorems) Exercise 8 (B) [Pages 107 - 112]
Multiple Choice Type: Choose the correct answer from the options given below.
For the polynomial x5 – x4 + x3 – 8x2 + 6x + 15, the maximum number of linear factors is ______.
9
6
7
5
If f(x) = 3x + 8; the value of f(x) + f(– x) is ______.
8
16
– 8
– 16
If x25 + x24 is divided by (x + 1), the result is ______.
49
1
0
–1
Factors of 3x3 – 2x2 – 8x are ______.
x(3x2 – 2x – 8)
x(x – 2)(3x + 4)
2x(x – 4)(2x + 1)
x(x – 4)(2x + 1)
Factors of 4 + 4x – x2 – x3 are ______.
(2 + x)(2 – x)(1 + x)
(x – 2)(1 + x)(2 + x)
(x + 2)(x – 2)(1 – x)
(2 + x)(x – 1)(2 – x)
Using the Factor Theorem, show that (x – 2) is a factor of x3 – 2x2 – 9x + 18. Hence, factorise the expression x3 – 2x2 – 9x + 18 completely.
Using the Factor Theorem, show that (x + 5) is a factor of 2x3 + 5x2 – 28x – 15. Hence, factorise the expression 2x3 + 5x2 – 28x – 15 completely.
Using the Remainder Theorem, factorise the following completely:
3x3 + 2x2 – 19x + 6
Using the Remainder Theorem, factorise the following completely:
2x3 + x2 – 13x + 6
Using the Remainder Theorem, factorise the following completely:
3x3 + 2x2 – 23x – 30
Using the Remainder Theorem, factorise the following completely:
4x3 + 7x2 – 36x – 63
Using the Remainder Theorem, factorise the following completely:
x3 + x2 – 4x – 4
Using the Remainder Theorem, factorise the expression 3x3 + 10x2 + x – 6. Hence, solve the equation 3x3 + 10x2 + x – 6 = 0
Factorise the expression f(x) = 2x3 – 7x2 – 3x + 18. Hence, find all possible values of x for which f(x) = 0.
Given that x – 2 and x + 1 are factors of f(x) = x3 + 3x2 + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).
The expression 4x3 – bx2 + x – c leaves remainders 0 and 30 when divided by x + 1 and 2x – 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely.
If x + a is a common factor of expressions f(x) = x2 + px + q and g(x) = x2 + mx + n; show that : `a = (n - q)/(m - p)`
Find the number that must be subtracted from the polynomial 3y3 + y2 – 22y + 15, so that the resulting polynomial is completely divisible by y + 3.
Selina solutions for Concise Mathematics [English] Class 10 ICSE 8 Factorization of Polynomials (Remainder and Factor Theorems) TEST YOURSELF [Pages 108 - 109]
Multiple Choice Type: Choose the correct answer from the options given below.
The remainder, when x3 – x2 + x – 1 is divided by x + 1, is ______.
0
– 4
2
4
The number of common factor(s) to two polynomials x2 − 9 and x3 − x2 − 9x + 9 is ______.
one
two
three
Is (x – 2) a factor of x3 – 4x2 – 11x + 30?
Yes
No
Nothing can be said
None of the above is true
4x2 – kx + 5 leaves a remainder 2 when divided by x – 1. The value of k is ______.
– 6
6
7
– 7
If mx2 – nx + 8 has x – 2 as a factor, then ______.
2m – n = 4
2m + n = 4
2n + m = 4
n – 2m = 4
Two polynomials x36 − 3x35 and x − 3.
Assertion (A): If x − 3 is factor of x36 − 3x35; the remainder is zero.
Reason (R): The polynomial x - a is factor of polynomial p(x) = x36 = ax35; if p(a) = 0)
A is true, R is false.
A is false, R is true.
Both A and R are true and R is the correct reason for A.
Both A and R are true and R is the incorrect reason for A.
The polynomial 3x3 + 8x2 − 15x + k and one of its factors as (x − 1)
Assertion (A): The value of k = 4
Reason (R): x − 1 = 0 ⇒ x = 1
∴ 3(1)3 + 8(1)2 − 15 × (1) + k = 0
A is true, R is false.
A is false, R is true.
Both A and R are true and R is the correct reason for A.
Both A and R are true and R is the incorrect reason for A.
The polynomial x2 + x + b has (x + 3) as a factor of it.
Statement (1): The value of b is −4 .
Statement (2): (x + 3) is a factor of x2 + x + b
⇒ (3)2 + 3 + b = 0
Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.
When x3 + 3x2 – mx + 4 is divided by x – 2, the remainder is m + 3. Find the value of m.
What should be subtracted from 3x3 – 8x2 + 4x – 3, so that the resulting expression has x + 2 as a factor?
If (x + 1) and (x – 2) are factors of x3 + (a + 1)x2 – (b – 2)x – 6, find the values of a and b. And then, factorise the given expression completely.
If x – 2 is a factor of x2 + ax + b and a + b = 1, find the values of a and b.
Find the value of ‘m’, if mx3 + 2x2 – 3 and x2 – mx + 4 leave the same remainder when each is divided by x – 2.
The polynomial px3 + 4x2 – 3x + q is completely divisible by x2 – 1; find the values of p and q. Also, for these values of p and q, factorize the given polynomial completely.
When the polynomial x3 + 2x2 – 5ax – 7 is divided by (x – 1), the remainder is A and when the polynomial x3 + ax2 – 12x + 16 is divided by (x + 2), the remainder is B. Find the value of ‘a’ if 2A + B = 0.
(3x + 5) is a factor of the polynomial (a – 1)x3 + (a + 1)x2 – (2a + 1)x – 15. Find the value of ‘a’, factorise the given polynomial completely.
Using remainder theorem, find the value of k if on dividing 2x3 + 3x2 – kx + 5 by x – 2, leaves a remainder 7.
Find the value of 'a' if x – a is a factor of the polynomial 3x3 + x2 – ax – 81.
While factorizing a given polynomial using the remainder and factor theorem, a student finds that (x + 3) is a factor of 2x3 – x2 – 5x – 2.
- Is the student’s solution correct in stating that (x + 3) is a factor of the given polynomial?
- Give a valid reason for your answer.
- factorize the given polynomial completely.
Solutions for 8: Factorization of Polynomials (Remainder and Factor Theorems)
Selina solutions for Concise Mathematics [English] Class 10 ICSE chapter 8 - Factorization of Polynomials (Remainder and Factor Theorems)
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Concepts covered in Concise Mathematics [English] Class 10 ICSE chapter 8 Factorization of Polynomials (Remainder and Factor Theorems) are Remainder Theorem, Factor Theorem, Applications of Factor Theorem, Function and Polynomial, Division Algorithm for Polynomials, Remainder Theorem, Factor Theorem, Applications of Factor Theorem, Function and Polynomial, Division Algorithm for Polynomials, Remainder Theorem, Factor Theorem, Applications of Factor Theorem, Function and Polynomial, Division Algorithm for Polynomials.
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