CBSE Class 10 Polynomials Sure Shot Questions Set 04

Polynomial in one variable

An algebraic expression of the form
\( p(x) = a_n x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \dots + a_1 x^1 + a_0 x^0 \), where
(i) \( a_n \neq 0 \)
(ii) \( a_0, a_1, a_2, \dots, a_n \) are real numbers
(iii) power of \( x \) is a positive integer, is called a polynomial in one variable.
Hence, \( a_n, a_{n-1}, a_{n-2}, \dots, a_0 \) are coefficients of \( x^n, x^{n-1}, \dots, x^0 \) respectively and \( a_n x^n, a_{n-1} x^{n-1}, a_{n-2} x^{n-2}, \dots \) are terms of the polynomial. Here the term \( a_n x^n \) is called the leading term and its coefficient \( a_n \), the leading coefficient.

Degree of polynomials

Degree of the polynomial in one variable is the largest exponent of the variable. For example, the degree of the polynomial \( 3x^7 - 4x^6 + x + 9 \) is 7 and the degree of the polynomial \( 5x^6 - 4x^2 - 6 \) is 6.

Classification of polynomials

Polynomials classified by degree

• Zero polynomial: Degree is \( -\infty \) (undefined); General form: 0; Example: 0.
• (Non-zero) constant polynomial: Degree: 0; General form: \( a; (a \neq 0) \); Example: 1.
• Linear polynomial: Degree: 1; General form: \( ax + b; (a \neq 0) \); Example: \( x + 1 \).
• Quadratic polynomial: Degree: 2; General form: \( ax^2 + bx + c; (a \neq 0) \); Example: \( x^2 + 1 \).
• Cubic polynomial: Degree: 3; General form: \( ax^3 + bx^2 + cx + d; (a \neq 0) \); Example: \( x^3 + 1 \).

A polynomial of degree n, for n greater than 3, is called a polynomial of degree n.

Polynomials classified by terms

• Monomials: Polynomials having only one term are called monomials. E.g. \( 2, 2x, 7y^5, 12t^7 \) etc.
• Binomials: Polynomials having exactly two terms are called binomials. E.g. \( p(x) = 2x + 1, r(y) = 2y^7 + 5y^6 \). etc.
• Trinomials: Polynomials having exactly three terms are called trinomials. E.g. \( p(x) = 2x^2 + x + 6, q(y) = 9y^6 + 4y^2 + 1 \) etc.

Zeros / Roots of a polynomial / equation

The value of the variable x, for which the polynomial \( f(x) \) becomes zero is called zero of the polynomial.
E.g. : consider, a polynomial \( p(x) = x^2 - 5x + 6 \); replace x by 2 and 3.
\( p(2) = (2)^2 - 5 \times 2 + 6 = 4 - 10 + 6 = 0 \),
\( p(3) = (3)^2 - 5 \times 3 + 6 = 9 - 15 + 6 = 0 \)
\(\therefore\) 2 and 3 are the zeros of the polynomial \( p(x) \).

Roots of a polynomial equation

An expression \( f(x) = 0 \) is called a polynomial equation if \( f(x) \) is a polynomial of degree \( n \geq 1 \).
A real number \( \alpha \) is a root of a polynomial \( f(x) = 0 \) if \( f(\alpha) = 0 \) i.e. \( \alpha \) is a zero of the polynomial \( f(x) \).
E.g. consider the polynomial \( f(x) = 3x - 2 \), then \( 3x - 2 = 0 \) is the corresponding polynomial equation.
Here, \( f\left(\frac{2}{3}\right) = 3\left(\frac{2}{3}\right) - 2 = 0 \)
i.e. \( \frac{2}{3} \) is a zero of the polynomial \( f(x) = 3x - 2 \)
or \( \frac{2}{3} \) is a root of the polynomial equation \( 3x - 2 = 0 \)

Important concepts

• A non-zero constant is a polynomial of degree zero, but the degree of zero polynomial is not defined.
• If the sum of the co-efficients of polynomial is zero, then \( (x - 1) \) is a factor of the polynomial.
• A polynomial in x is said to be a polynomial in standard form, if the powers of x are either in ascending order or in descending order.
• A polynomial of degree \( n \geq 1 \) can have at the most n real zeros.
• A non-zero constant polynomial has no zero.
• Every linear polynomial has one and only one zero.
• A quadratic polynomial \( ax^2 + bx + c, a \neq 0 \) can have at most two real zeros. In some cases, it may not have any real zero.
• Zero of a polynomial is actually the solution of the curve, \( y = f(x) \) and the line \( y = 0 \).

Remainder theorem

Statement : Let \( p(x) \) be a polynomial of degree \( \geq 1 \) and 'a' is any real number. If \( p(x) \) is divided by \( (x - a) \), then the remainder is \( p(a) \).

• \( p(-a) \) is remainder on dividing \( p(x) \) by \( (x + a) \) [\(\because x + a = 0 \implies x = -a\)]
• \( p\left(\frac{b}{a}\right) \) is remainder on dividing \( p(x) \) by \( (ax - b) \) [\(\because ax - b = 0 \implies x = b/a\)]
• \( p\left(-\frac{b}{a}\right) \) is remainder on dividing \( p(x) \) by \( (ax + b) \) [\(\because ax + b = 0 \implies x = -b/a\)]
• \( p\left(\frac{b}{a}\right) \) is remainder on dividing \( p(x) \) by \( (b - ax) \) [\(\because b - ax = 0 \implies x = b/a\)]

Factor theorem

Statement : Let \( f(x) \) be a polynomial of degree \( \geq 1 \) and a be any real constant such that \( f(a) = 0 \), then \( (x - a) \) is a factor of \( f(x) \). Conversely, if \( (x - a) \) is a factor of \( f(x) \), then \( f(a) = 0 \).

\( P(x) \) is a polynomial of degree \( \geq 1 \) and “a” is a real number then
\( p(a) = 0 \implies (x - a) \) is a factor of \( p(x) \)

• \( (x - a) \) is a factor of \( p(x) \) then \( p(a) = 0 \)
• \( ax - b \) is a factor of \( p(x) \) then \( P\left(\frac{b}{a}\right) = 0 \)
• \( ax + b \) is a factor of \( p(x) \) then \( P\left(-\frac{b}{a}\right) = 0 \)
• \( (x - a) \) is a factor of \( x^n - a^n \) where “n” is an odd positive integer
• \( (x + a) \) is a factor of \( x^n - a^n \) where “n” is an odd positive integer
• \( (x + a) \) is factor of \( x^n - a^n \) where “n” is positive even integer.
• \( (x^n + a^n) \) is not divisible by \( x + a \) when “n” is even
• \( x^n + a^n \) is not divisible by \( x - a \) for any “n”
• If \( x - 1 \) is a factor of polynomial of degree ‘n’ then the condition is sum of the coeffecients is zero.
• If \( (x + 1) \) is a factor of polynomial of degree ‘n’ then the condition is sum of the coefficients of even terms is equal to the coefficients of odd terms.

Relationship between the zeros and coefficients of a polynomial

For a linear polynomial \( ax + b, (a \neq 0) \), we have,
zero of a linear polynomial \( = -\frac{b}{a} = -\frac{(\text{constant term})}{(\text{coefficient of } x)} \)

For a quadratic polynomial \( ax^2 + bx + c (a \neq 0) \), with \( \alpha \) and \( \beta \) as it's zeros, \( (x - \alpha) \) and \( (x - \beta) \) are the factors of \( ax^2 + bx + c \).
Therefore, \( ax^2 + bx + c = K(x - \alpha) (x - \beta) \), (where K is a constant to balance the equation of the coefficient of \( x^2 \) i.e. \( a \neq 1 \).)
\( = K x^2 - K (\alpha + \beta) x + K \alpha \beta \)
comparing the coefficients of \( x^2, x \) and constant terms on both the sides, we get
\( a = K, b = -K (\alpha + \beta) \) and \( c = K \alpha \beta \)

This gives
Sum of zeros \( = \alpha + \beta = -\frac{b}{a} = -\frac{(\text{coefficient of } x)}{(\text{coefficient of } x^2)} \)
Product of zeros \( = \alpha \beta = \frac{c}{a} = \frac{(\text{constant term})}{(\text{coefficient of } x^2)} \)

If \( \alpha \) and \( \beta \) are the zeros of a quadratic polynomial \( f(x) \). Then polynomial \( f(x) \) is given by
\( f(x) = K\{x^2 - (\alpha + \beta)x + \alpha \beta\} \)
or \( f(x) = K\{x^2 - (\text{sum of the zeros}) x + \text{product of the zeros}\} \)
where K is a constant.

Symmetric functions of the zeros

Let \( \alpha, \beta \) be the zeros of a quadratic polynomial, then the expression of the form \( \alpha + \beta; (\alpha^2 + \beta^2); \alpha \beta \) are called the functions of the zeros. By symmetric function we mean that the function remain invariant (unaltered) in values when the roots are changed cyclically. In other words, an expression involving \( \alpha \) and \( \beta \) which remains unchanged by interchanging \( \alpha \) and \( \beta \) is called a symmetric function of \( \alpha \) and \( \beta \).
Some useful relations involving \( \alpha \) and \( \beta \) are :-

• \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \)
• \( (\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha \beta \)
• \( \alpha^2 - \beta^2 = (\alpha + \beta) (\alpha - \beta) = (\alpha + \beta) \sqrt{(\alpha + \beta)^2 - 4\alpha \beta} \)
• \( \alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha \beta (\alpha + \beta) \)
• \( \alpha^3 - \beta^3 = (\alpha - \beta)^3 + 3\alpha \beta (\alpha - \beta) \)
• \( \alpha^4 - \beta^4 = (\alpha^2 + \beta^2) (\alpha + \beta) (\alpha - \beta) = [(\alpha + \beta)^2 - 2\alpha \beta] (\alpha + \beta) \sqrt{(\alpha + \beta)^2 - 4\alpha \beta} \)
• \( \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha \beta)^2 = [(\alpha + \beta)^2 - 2\alpha \beta]^2 - 2(\alpha \beta)^2 \)
• \( \alpha^5 + \beta^5 = (\alpha^3 + \beta^3) (\alpha^2 + \beta^2) - \alpha^2 \beta^2 (\alpha + \beta) = [(\alpha + \beta)^3 - 3\alpha \beta (\alpha + \beta)] [(\alpha + \beta)^2 - 2\alpha \beta] - (\alpha \beta)^2 (\alpha + \beta) \)

Division algorithm for polynomials

If \( f(x) \) is a polynomial and \( g(x) \) is a non-zero polynomial, then there exist two polynomials \( q(x) \) and \( r(x) \) such that
\( f(x) = g(x) \times q(x) + r(x) \), where \( r(x) = 0 \) or degree \( r(x) < \) degree \( g(x) \). In other words,
Dividend = Divisor \(\times\) Quotient + Remainder
Remark : If \( r(x) = 0 \), then polynomial \( g(x) \) is a factor of polynomial \( f(x) \).

Algebraic identities

An algebraic identity is an algebraic equation that is true for all values of the variables present in the equation.

• \( (x + y)^2 = x^2 + 2xy + y^2 \)
• \( (x - y)^2 = x^2 - 2xy + y^2 \)
• \( x^2 - y^2 = (x + y) (x - y) \)
• \( (x + a) (x + b) = x^2 + (a + b) x + ab \)
• \( (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx \)
• \( (x + y)^3 = x^3 + y^3 + 3xy (x + y) \)
• \( (x - y)^3 = x^3 - y^3 - 3xy (x - y) \)
• \( x^3 + y^3 = (x + y) (x^2 - xy + y^2) \)
• \( x^3 - y^3 = (x - y) (x^2 + xy + y^2) \)
• \( x^3 + y^3 + z^3 - 3xyz = (x + y + z) (x^2 + y^2 + z^2 - xy - yz - zx) \)
• If \( a + b + c = 0 \), then \( a^3 + b^3 + c^3 = 3abc \)

Question. If \( x + \frac{1}{x} = 5 \), then the value of \( x^3 + \frac{1}{x^3} \) is
(a) 110
(b) 90
(c) 80
(d) 50
Answer: (a) 110

Question. If \( x^3 - (x + 1)^2 = 2001 \) then the value of \( x \) is
(a) 14
(b) 13
(c) 10
(d) None
Answer: (b) 13

Question. The square root of \( \frac{x^2}{y^2} + \frac{y^2}{4x^2} - \frac{x}{y} + \frac{y}{2x} - \frac{3}{4} \) is
(a) \( \frac{x}{y} - \frac{1}{2} - \frac{y}{2x} \)
(b) \( \frac{x}{y} + \frac{1}{2} - \frac{y}{2x} \)
(c) \( \frac{x}{y} + \frac{1}{2} + \frac{y}{2x} \)
(d) \( \frac{x}{y} - \frac{1}{4} - \frac{y}{2x} \)
Answer: (a) \( \frac{x}{y} - \frac{1}{2} - \frac{y}{2x} \)

Question. If the zeros of the polynomial \( ax^2 + bx + c \) be in the ratio \( m : n \), then
(a) \( b^2 mn = (m^2 + n^2) ac \)
(b) \( (m + n)^2 ac = b^2 mn \)
(c) \( b^2 (m^2 + n^2) = mnac \)
(d) None of these
Answer: (b) \( (m + n)^2 ac = b^2 mn \)

Question. If \( \alpha \neq \beta \) and the difference between the roots of the polynomials \( x^2 + ax + b \) and \( x^2 + bx + a \) is the same, then
(a) \( a + b + 4 = 0 \)
(b) \( a + b - 4 = 0 \)
(c) \( a - b + 4 = 0 \)
(d) \( a - b - 4 = 0 \)
Answer: (a) \( a + b + 4 = 0 \)

Question. If \( \alpha \neq \beta \) and \( \alpha^2 = 5\alpha - 3, \beta^2 = 5\beta - 3 \), then the polynomial whose zeros are \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \) is :
(a) \( 3x^2 - 25x + 3 \)
(b) \( x^2 - 5x + 3 \)
(c) \( x^2 + 5x - 3 \)
(d) \( 3x^2 - 19x + 3 \)
Answer: (d) \( 3x^2 - 19x + 3 \)

Question. The factors of \( a^2(b^3 - c^3) + b^2(c^3 - a^3) + c^2(a^3 - b^3) \) are
(a) \( (a - b) (b - c) (c - a) (ab + bc + ca) \)
(b) \( (a + b) (b + c) (c + a) (ab + bc + ca) \)
(c) \( (a - b) (b - c) (c - a) (ab - bc - ca) \)
(d) None of these
Answer: (a) \( (a - b) (b - c) (c - a) (ab + bc + ca) \)

Question. If \( p, q \) are zeros of \( x^2 + px + q \), then
(a) \( p = 1 \)
(b) \( p = 1 \) or \( 0 \)
(c) \( p = -2 \)
(d) \( p = -2 \) or \( 0 \)
Answer: (b) \( p = 1 \) or \( 0 \)

Question. On simplifying \( (a + b)^3 + (a - b)^3 + 6a(a^2 - b^2) \) we get
(a) \( 8a^2 \)
(b) \( 8a^2b \)
(c) \( 8a^3b \)
(d) \( 8a^3 \)
Answer: (d) \( 8a^3 \)

Question. Factors of \( (42 - x - x^2) \) are
(a) \( (x - 7)(x - 6) \)
(b) \( (x + 7)(x - 6) \)
(c) \( (x + 7)(6 - x) \)
(d) \( (x + 7)(x + 6) \)
Answer: (c) \( (x + 7)(6 - x) \)

Question. Factors of \( x^2 + \frac{x}{6} - \frac{1}{6} \) are
(a) \( \frac{1}{6}(2x+1)(3x+1) \)
(b) \( \frac{1}{6}(2x+1)(3x-1) \)
(c) \( \frac{1}{6}(2x-1)(3x-1) \)
(d) \( \frac{1}{6}(2x-1)(3x+1) \)
Answer: (b) \( \frac{1}{6}(2x+1)(3x-1) \)

Question. Value of \( \frac{a^3 + b^3 + c^3 - 3abc}{ab + bc + ca - a^2 - b^2 - c^2} \), when \( a = -5, b = -6, c = 10 \) is
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (a) 1

Question. If \( (x + y + z) = 1, xy + yz + zx = -1, xyz = -1 \), then the value of \( x^3 + y^3 + z^3 \) is
(a) -1
(b) 1
(c) 2
(d) -2
Answer: (b) 1

Question. In method of factorization of an algebraic expression, Which of the following statements is false?
(a) Taking out a common factor from two or more terms
(b) Taking out a common factor from a group of terms
(c) By using remainder theorem
(d) By using standard identities
Answer: (c) By using remainder theorem

Question. Factors of \( (a + b)^3 - (a - b)^3 \) are
(a) \( 2ab(3a^2 + b^2) \)
(b) \( ab(3a^2 + b^2) \)
(c) \( 2b(3a^2 + b^2) \)
(d) \( 3a^2 + b^{20} \)
Answer: (c) \( 2b(3a^2 + b^2) \)

Question. The homogeneous function of the second degree in x and y having \( 2x - y \) as a factor, taking the value 2 when \( x = y = 1 \) and vanishing if \( x = -1, y = 1 \) is
(a) \( 2x^2 + xy - y^2 \)
(b) \( 3x^2 - 2xy + y^2 \)
(c) \( x^2 + xy - 2y^2 \)
(d) None of these
Answer: (a) \( 2x^2 + xy - y^2 \)

Question. The common quantity that must be added to each term of \( a^2 : b^2 \) to make it equal to \( a : b \) is
(a) \( ab \)
(b) \( a + b \)
(c) \( a - b \)
(d) \( \frac{a}{b} \)
Answer: (a) \( ab \)

Question. If the polynomial \( 16x^4 - 24x^3 + 41x^2 - mx + 16 \) be a perfect square, then the value of "m" is
(a) 12
(b) -12
(c) 24
(d) -24
Answer: (c) 24

Question. If \( a - b = 3, a + b + x = 2 \), then the value of \( (a - b)[x^3 - 2ax^2 + a^2x - (a + b)b^2] \) is
(a) 84
(b) 48
(c) 32
(d) 36
Answer: (b) 48

Question. If \( abx^2 = (a - b)^2(x + 1) \), then the value of \( 1 + \frac{4}{x} + \frac{4}{x^2} \) is:-
(a) \( \left(\frac{a-b}{a+b}\right)^2 \)
(b) \( \left(\frac{a+b}{a-b}\right)^2 \)
(c) \( \left(\frac{a}{a+b}\right)^2 \)
(d) \( \left(\frac{b}{a+b}\right)^2 \)
Answer: (b) \( \left(\frac{a+b}{a-b}\right)^2 \)

Question. Let \( \alpha, \beta \) be the zeros of the polynomial \( (x - a)(x - b) - c \) with \( c \neq 0 \). Then the zeros of the polynomial \( (x - \alpha) (x - \beta) + c \) are
(a) a, c
(b) b, c
(c) a, b
(d) \( a + c, b + c \)
Answer: (c) a, b

Question. A homogeneous expression of second degree in x & y is
(a) \( ax^2 + bx + c \)
(b) \( ax^2 + bx + cy \)
(c) \( ax^2 + bx + cy^2 \)
(d) \( ax^2 + bxy + cy^2 \)
Answer: (d) \( ax^2 + bxy + cy^2 \)

Question. If the sum of the zeros of the polynomial \( x^2 + px + q \) is equal to the sum of their squares, then
(a) \( p^2 - q^2 = 0 \)
(b) \( p^2 + q^2 = 2q \)
(c) \( p^2 + p = 2q \)
(d) None of these
Answer: (c) \( p^2 + p = 2q \)

Question. The G.C.D of \( x^2 - 3x + 2 \) and \( x^2 - 4x + 4 \) is
(a) \( x - 2 \)
(b) \( (x - 2)(x - 1) \)
(c) \( (x - 2)^2 \)
(d) \( (x - 2)^3(x - 1) \)
Answer: (a) \( x - 2 \)

Question. The L.C.M. of \( 22x(x + 1)^2 \) and \( 36x^2(2x^2 + 3x + 1) \) is
(a) \( 2x(x + 1) \)
(b) \( 396x^2(x + 1)^2(2x + 1) \)
(c) \( 792x^3(x + 1)^2 (2x^2 + 3x + 1) \)
(d) None of these
Answer: (b) \( 396x^2(x + 1)^2(2x + 1) \)

Question. The L.C.M of \( x^3 - 8 \) and \( x^2 - 5x + 6 \) is
(a) \( x - 2 \)
(b) \( x^2 + 2x + 4 \)
(c) \( (x - 2)(x^2 + 2x + 4) \)
(d) \( (x - 2)(x - 3)(x^2 + 2x + 4) \)
Answer: (d) \( (x - 2)(x - 3)(x^2 + 2x + 4) \)

Question. If the G.C.D. of the polynomials \( x^3 - 3x^2 + px + 24 \) and \( x^2 - 7x + q \) is \( (x - 2) \), then the value of \( (p + q) \) is:
(a) 0
(b) 20
(c) -20
(d) 40
Answer: (a) 0

Question. If the L.C.M. of two polynomials p(x) and q(x) is \( (x + 3)(x - 2)^2(x - 6) \) and their H.C.F. is \( (x - 2) \). If \( p(x) = (x + 3)(x - 2)^2 \), then \( q(x) = \)
(a) \( (x + 3)(x - 2) \)
(b) \( x^2 - 3x - 18 \)
(c) \( x^2 - 8x + 12 \)
(d) none of these
Answer: (c) \( x^2 - 8x + 12 \)

Question. The G.C.D. of two polynomials is \( (x - 1) \) and their L.C.M. is \( x^6 - 1 \). If one of the polynomials is \( x^3 - 1 \), then the other polynomial is_____.
(a) \( x^3 - 1 \)
(b) \( x^4 - x^3 + x - 1 \)
(c) \( x^2 - x + 1 \)
(d) None of these
Answer: (b) \( x^4 - x^3 + x - 1 \)

Question. The L.C.M. of 2x and 8 is
(a) 2x
(b) 4x
(c) 8x
(d) 16x
Answer: (c) 8x

Question. If \( x^2 + \frac{1}{x^2} = 38 \), then the value of \( x - \frac{1}{x} \) is
(a) 6
(b) 4
(c) 0
(d) None
Answer: (a) 6

Question. The simplest form of \( (2x + 3)^3 - (2x - 3)^3 \) is
(a) \( 54 + 72x^2 \)
(b) \( 72 + 54x^2 \)
(c) \( 54 + 54x^2 \)
(d) None of these
Answer: (b) \( 72 + 54x^2 \)

Question. The simplest form of \( (p - q)^3 + (q - r)^3 + (r - p)^3 \) is
(a) \( 4(p - q)(q - r)(r - p) \)
(b) \( 2(p - q)(q - r)(r - p) \)
(c) \( 3(p - q)(q - r)(r - p) \)
(d) None of these
Answer: (c) \( 3(p - q)(q - r)(r - p) \)

Question. The square root of \( x^4 + 6x^3 + 17x^2 + 24x + 16 \) is
(a) \( x^2 + 3x + 4 \)
(b) \( 2x^2 + 3x + 4 \)
(c) \( 3x^2 + 3x + 4 \)
(d) None of these
Answer: (a) \( x^2 + 3x + 4 \)

Question. The square root of \( x^4 - 2x^3 + 3x^2 - 2x + 1 \) is
(a) \( x^2 + x + 1 \)
(b) \( x^2 - x + 1 \)
(c) \( x^2 + x - 1 \)
(d) \( x^2 - x - 1 \)
Answer: (b) \( x^2 - x + 1 \)

Question. The value of \( \lambda \) for which one zero of \( 3x^2 - (1 + 4\lambda) x + \lambda^2 + 2 \) may be one-third of the other is
(a) 4
(b) \( \frac{33}{8} \)
(c) \( \frac{17}{4} \)
(d) \( \frac{31}{8} \)
Answer: (d) \( \frac{31}{8} \)

Question. The factors of \( a^3(b - c) + b^3(c - a) + c^3(a - b) \) are
(a) \( (a + b + c) (a - b) (b - c) (c - a) \)
(b) \( - (a + b + c) (a - b) (b - c) (c - a) \)
(c) \( 2 (a + b + c) (a - b) (b - c) (c - a) \)
(d) \( - 2 (a + b + c) (a - b) (b - c) (c - a) \)
Answer: (b) \( - (a + b + c) (a - b) (b - c) (c - a) \)

Question. The value of 'a', for which one root of the quadratic polynomial \( (a^2 - 5a + 3) x^2 + (3a - 1) x + 2 \) is twice as large as the other, is
(a) \( - \frac{1}{3} \)
(b) \( \frac{2}{3} \)
(c) \( - \frac{2}{3} \)
(d) \( \frac{1}{3} \)
Answer: (b) \( \frac{2}{3} \)

Question. If the polynomial \( x^{19} + x^{17} + x^{13} + x^{11} + x^7 + x^5 + x^3 \) is divided by \( (x^2 + 1) \), then the remainder is
(a) 1
(b) \( x^2 + 4 \)
(c) -x
(d) x
Answer: (c) -x

Question. If \( (x - 2) \) is a common factor of \( x^3 - 4x^2 + ax + b \) and \( x^3 - ax^2 + bx + 8 \), then the values of a and b are respectively
(a) 3 and 5
(b) 2 and -4
(c) 4 and 0
(d) 0 and 4
Answer: (c) 4 and 0

Question. If the expressions \( ax^3 + 3x^2 - 3 \) and \( 2x^3 - 5x + a \) on dividing by \( x - 4 \) leave the same remainder, then the value of a is
(a) 1
(b) 0
(c) 2
(d) -1
Answer: (a) 1

Question. If the polynomial \( x^6 + px^5 + qx^4 - x^2 - x - 3 \) is divisible by \( x^4 - 1 \), then the value of \( p^2 + q^2 \) is
(a) 1
(b) 5
(c) 10
(d) 13
Answer: (c) 10

Question. If \( 3x^3 + 2x^2 - 3x + 4 = (Ax + B)(x - 1)(x + 2) + C(x - 1) + D \) for all values of x, then \( A + B + C + D \) is
(a) 0
(b) 14
(c) 10
(d) All
Answer: (b) 14

Question. The expression \( x^3 + gx^2 + hx + k \) is divisible by both x and \( x - 2 \) but leaves a remainder of 24 when divided by \( x + 2 \) then the values of g, h and k are
(a) g = 10, h = -3, k = 0
(b) g = 3, h = -10, k = 0
(c) g = 10, h = -2, k = 3
(d) None of these
Answer: (b) g = 3, h = -10, k = 0

Question. The value of m if \( 2x^m + x^3 - 3x^2 - 26 \) leaves a remainder of 226 when it is divided by \( x - 2 \).
(a) 0
(b) 7
(c) 10
(d) All of these
Answer: (b) 7

Question. The expression \( Ax^3 + x^2 + Bx + C \) leaves remainder of \( \frac{21}{4} \) when divided by \( 1 - 2x \) and 18 when divided by x. Given also the expression has a factor of \( (x - 2) \), the values of A, B and C are
(a) A = 5, B = -9, C = 3
(b) A = 27, B = -18, C = 4
(c) A = 4, B = -27, C = 18
(d) None of these
Answer: (c) A = 4, B = -27, C = 18

Question. If \( h(x) = 2x^3 + (6a^2 - 10) x^2 + (6a + 2) x - 14a - 2 \) is exactly divisible by \( x - 1 \) but not by \( x + 1 \), then the value of a is
(a) 0
(b) -1
(c) 10
(d) 2
Answer: (d) 2

Question. Given the polynomial is exactly divided by \( x + 1 \), and when it is divided by \( 3x - 1 \), the remainder is 4. The polynomial gives a remainder \( hx + k \) when divided by \( 3x^2 + 2x - 1 \) then the values of h and k are
(a) h = 2, k = 3
(b) h = 3, k = 3
(c) h = 3, k = 2
(d) None of these
Answer: (b) h = 3, k = 3

Question. The remainder when \( f(x) = (x^4 - x^3 + 2x - 3) g(x) \) is divided by \( x - 3 \), given that \( x - 3 \) is a factor of \( g(x) + 3 \), where g(x) is a polynomial is
(a) 0
(b) -171
(c) 10
(d) 2
Answer: (b) -171

Question. If \( x^3 - hx^2 + kx - 9 \) has a factor of \( x^2 + 3 \), then the values of h and k are
(a) h = 3, k = 3
(b) h = 2, k = 2
(c) h = 2, k = 1
(d) None of these
Answer: (a) h = 3, k = 3