Class X ICSE Maths Chapter Remainder & Factor Theorem Exercise 6
Welcome to our comprehensive collection of questions on the topic "Remainder & Factor Theorem," crafted to help you build a solid understanding of these essential algebraic concepts. Our question bank covers a range of problems that apply the Remainder Theorem to find the remainder when a polynomial is divided by a linear divisor, and the Factor Theorem to determine the factors of polynomials. These questions are ideal for students looking to strengthen their skills in polynomial division and factorization. Whether you're preparing for exams or simply improving your algebraic knowledge, our collection is the perfect resource to practice and master the "Remainder & Factor Theorem."
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Q1 Find the remainder (without division) on dividing f(x) by (x - 2) where
(i) f(x) = 5x2 - 7x + 4
(ii) f(x) = 2x3 - 7x2 + 3,
Q2 Using remainder theorem, find the remainder on dividing f(x) by (x + 3) where
(i) f(x) = 2x2 - 5x + 1
(ii) f(x) = 3x3 + 7x2 - 5x + 1,
Q3 Find the remainder (without division) on dividing f(x) by (2x + 1) where
(i) f(x) = 4x2 + 5x + 3
(ii) f(x) = 3x3 - 7x2 + 4x + 11.
Q4 Using remainder theorem, find the value of k if on dividing 2x3 + 3x2 - kx + 5 by x - 2, leaves a remainder 7.
Q5 Using remainder theorem, find the value of a if the division of x3 + 5x2 - ax + 6 by (x — 1) leaves the remainder 2a.
Q6 (i) What number must be subtracted from 2x2 - 5x so that the resulting polynomial leaves the remainder 2 when divided by 2x + 1?
(ii) What number must be added to 2x3 - 7x2 + 2x so that the resulting polynomial leaves the remainder —2 when divided by 2x - 3?
Q7 (i) When divided by x - 3 the polynomials x3 - px2 + x +6 and : 2x3 - x3 - (p + 3) x — 6 leave the same remainder. Find the value of ‘p’.
(ii) Find ‘a’ if the two polynomials ax3 + 3x2 — 9 and 2x3 + 4x + a, leaves the same remainder when divided by x + 3.
(iii) The polynomials ax3 + 3x2 — 3 and 2x3 — 5x + q when divided by x - 4 leave the remainder r1 and r2 respectively. If 2r1 = r2 then find the value of a.
Q8 Using the remainder theorem, find the remainders obtained when x3 + (kx + 8) x + k is divided by x + 1 and x - 2. Hence, find k if the sum of two remainders is 1.
Q10 Without actual division, prove that x4 + 2x3 — 2x2 + 2x - 3 is exactly divisible by x2 + 2x - 3.
Q12 Using the factor theorem, show that (x — 2) is a factor of x3 + x2 - 4x — 4. Hence, factorise the polynomial completely.
Q13 Show that 2x + 7 is a factor of 2x3 + 5x2 - 11x - 14. Hence, factorise the given expression completely, using the factor theorem.
Q14 Use factor theorem to factorise the following polynomials completely :
(i) x3 + 2x2 - 5x - 6
(ii) x3 - 13x - 12.
Q15 Use Remainder Theorem to factorise the following polynomials completely:
(i) 2x3 + x2 - 13x + 6
(ii) 3x3 + 2x2 — 19x + 6
(iii) 2x3 + 3x2 - 9x - 10
(iv) x3 + 10x2 - 37x + 26.
Q18 If (x - 2) is a factor of 2x3 — x2 — px - 2, then
(i) find the value of p.
(ii) with this value of p, factorise the above expression completely.
Q19 What number should be subtracted from 2x3 — 5x2 + 5x so that the resulting polynomial has 2x — 3 as a factor?
Q20 (i) Find the value of the constants a and b, if (x — 2) and (x + 3) are both factors of the expression x3 + ax2 + bx — 12.
(ii) If (x + 2) and (x + 3) are factors of x3 + ax + b, find the values of a and b.
Q21 If (x + 2) and (x — 3) are factors of x3 + ax + b, find the values of a and b. With these values of a and b, factorise the given expression.
Q22 (x - 2) is a factor of the expression x3 + ax2 + bx + 6. When this expression is divided by (x — 3), it leaves the remainder 3. Find the values of a and b.
Q23 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.
Q24 If ax3 + 3x2 + bx — 3 has a factor (2x + 3) and leaves remainder -3 when divided by (x + 2), find the values of a and b. With these values of a and b, factorise the given expression.
Q25 Given f(x) = ax2 + bx + 2 and g(x) = bx2 + ax + 1. If x — 2 is a factor of f(x) but leaves the remainder —15 when it divides g(x), find the values of a and b. With these values of a and b, factorise the expression
f(x) + g(x) + 4x2 + 7x.
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