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Published on: 02/03/2019
Polynomials Important Questions
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Questions + Answers key
Take MCQ Mathematics Test
1.
Find the product: (x+y+2z)(x2+y2+4z2-xy-2yz-2zx).
2.
Factorize: a6-b6
3.
If \(x^2-3x+2\) is a factor of \(x^4-ax^2+b\) then find a and b
4.
Factorise each of the following: \(8a^3-b^3-12a^2b+6ab^2\)
5.
Evaluate the following products without multiplying directly: 95 x 96
6.
Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases: \(p(x)=2{ x }^{ 3 }+{ x }^{ 2 }-2x-1\ g(x)=x+1\)
7.
Verify whether the following are zeroes of the polynomial, indicated against them.
\(p(x)=lx+m,x=-\frac { m }{ l } \)
8.
Find p(0), p(1) and p(2) for each of the following polynomials:
p(y) = y2 - y + 1
9.
Classify the following as linear, quadratic and cubic polynomials:
x2 + x
10.
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
\(y+\frac { 2 }{ y } \)
11.
Factorize: (x2-3x)2-8(x2-3x)-20
12.
If \(x+\frac{1}{x}=7,\) then find the value of \(x^3+\frac{1}{x^3}\)
13.
If \(a^2+b^2+c^2=280\) and \(ab+bc+ca=\frac{9}{2},\)Then find the value of \((a+b+c)^3\)
14.
Find the value of the polynomial \(p(z)={ 3z }^{ 2 }=4z+\sqrt { 17 } \) when z=3.
15.
Write the various coefficients in the following polynomials:
7
16.
If \({ \left( \frac { 8 }{ 15 } \right) }^{ 3 }-{ \left( \frac { 1 }{ 3 } \right) }^{ 3 }-{ \left( \frac { 1 }{ 5 } \right) }^{ 3 }=\frac { x }{ 75 } \), find x.
17.
Factorize: x4-y4.
18.
Factorize: 12(x2+7)2-8(x2+7)(2x-1)-15(2x-1)2.
19.
Expand using suitable identity (2x-3y+z)2
20.
Find the remainder when x3+x2+x+1 is divided by x-\(\frac { 1 }{ 2 } \),using remainder theorem.
21.
If \(x+\frac{1}{x}=3\) then find \(x^3+\frac{1}{x^3}\)
22.
If \(x+y+z=1,\quad xy+yz+zx=-1 \)and xyz=-1 find the value of \(x^3+y^3+z^3\)
23.
Factorise: \((5a+\frac{2}{3})^2-(2a-\frac{1}{3})^2\)
24.
Factorise:
(i) \(49a^2+70ab+25b^2\)
(ii) \(\frac{25}{4}x^2-\frac{y^2}{9}.\)
25.
Factorise by using factor theorem:\(x^3+13x^2+32x+20\)
26.
If 2x-1 is a factor of \(4x^3-16x^2+10x+k\) then find the value of k.
27.
Find the remainder when the polynomial \(p(y)=y^4-3y^2+7y-10\) and (y-2).
28.
Divide p(x) by g(x), where p(x) = x + 3x2 – 1 and g(x) = 1 + x.
29.
Point out which of the following polynomials are monomials, binomials or trinomials?
\({ 7x }^{ 2 }-5\)
30.
Write the coefficient of x3 of the following polynomials:
\(4x+{ x }^{ 3 }+{ x }^{ 6 }-{ 7x }^{ 2 }\)
31.
Product of \((x-\frac{1}{x})(x+\frac{1}{x})(x^2+\frac{1}{x^2})\) is:
\(x^4+\frac{1}{x^4}\)
\(x^3+\frac{1}{x^3}-2\)
\(x^4-\frac{1}{x^4}\)
\(x^2+\frac{1}{x^2}+2\)
32.
If a+b+c=5 and ab+bc+ca=10, then the value of \(a^3+b^3+c^3-3abc\) is:
20
-25
10
5
33.
One of the factors of \(42+y-y^2\) is:
\((7+y)\)
\((6-y)\)
\((7-y)\)
\((-6+y)\)
34.
The factor of \((2a-b)^3+(b-2c)^3+8(c-a)^3\) is:
(2a-b)(b-2c)(c-a)
3(2a-b)(b-2c)(c-a)
6(6a-b)(b-2c)(c-a)
2a x b x 2c
35.
The value of p for which x+p is a factor of \(x^2+px+3-p \) is:
1
-1
3
-3
36.
If \(x^{11}+101 \) is divided by (x+1), the remainder is:
-1
102
0
100
37.
If a polynomial f(x) is divided by x-a, then remainder is:
f(0)
f(a)
f(-a)
f(a)-f(0)
38.
Find \(p(\frac{1}{3})\) for \(p(t)=t^2-t+2\)
\(\frac{22}{9}\)
\(\frac{14}{9}\)
\(\frac{16}{9}\)
\(\frac{15}{9}\)
39.
The zeros of \(f(x)=x^2+2x\) are:
0, -2
1, 2
0, 2
1, -2
40.
The coefficient of x2 in \((3x^2-5)(4+4x^2)\) is:
-12
5
-8
8
41.
The degree of the polynomial \(p(x)=(x-7)^3-x^3\) is:
3
2
1
0
42.
Degree of the polynomial \(4x^4+0x^3+0x^5+5x+7\) is:
7
5
4
3
43.
Which of the following is a binomial in y?
\(y^2+2\)
\(y+\frac{1}{y}+2\)
\(\sqrt{y}+\sqrt{2}y\)
\(y\sqrt{y}+1\)
44.
\(y+\frac{1}{y}\) is:
polynomial of degree 1
polynomial of degree 2
polynomial of degree 3
Not a polynomial
45.
The compact form of (x+y)(x-y) is
\((x+y)(x-y)=x^2-y^2\) is an algebraic identity
\(x^2+y^2\)
\(x^2-2xy+y^2\)
\(x^2+2xy+y^2\)
\(x^2-y^2\)
46.
Write the factors of polynomial 4x2+y2+4xy+8x+4y+4
47.
Factorize: 8y3-125x3.
48.
Factorize: 20x2-9x+1.
49.
Factorize: x2-3x.
50.
Write an example of a constant polynomial.
1.
(x+y+2z)(x2+y2+4z2-xy-2yz-2zx)
=(x+y+2z)(x2+y2+(2z)2-x\(\times\)y-y\(\times\)2z-x\(\times\)2z]
We know that
(a+b+c)(a2+b2+c2-ab-bc-ca)
=a3+b3+c3-3abc
\(\therefore\) (x+y+2z)(x2+y2+(2z)2-x\(\times\)y-y\(\times\)2z-x\(\times\)2z)
=(x)3+(y)3+(2z)3-3\(\times\)x\(\times\)y\(\times\)2z
= x3+y3+8z3-6xyz.
2.
a6-b6=(a3)2-(b3)2
=(a3-b3)(a3+b3)
=(a-b)(a2+b2+ab)(a+b)(a2+b2-ab)
=(a-b)(a+b)(a2+b2+ab)(a2+b2-ab).
3.
a=5, b=4
4.
\(8a^3-b^3-12a^2b+6ab^2\)
\(=(2a)^3-(b)^3-3(2a)(b)(2a-b)\)
\(=(2a-b)^3\) | Using Identity VII
\(=(2a-b)(2a-b)(2a-b)\)
5.
95 x 96
95 x 96=(90+5) x (90+6)
=(90)2+(5+6)(90)+(5)(6) | Using Identity IV
=8100+990+30=9120.
Aliter
95 x 96 = (100-5) x (100-4)
={100+(-5)} {100+(-4)}
=(100)2+{(-5)+(-4)}(100)+(-5)(-4) | Using Identity IV
=10000-900+20
=9120.
6.
\(p(x)=2{ x }^{ 3 }+{ x }^{ 2 }-2x-1\)
\(\\ g(x)=x+1\)
\(\\ g(x)=0\)
\(\Rightarrow \ x+1=0\ \Rightarrow \ x=-1\)
\(\therefore\) Zero of g(x) is -1.
Now, P(-1)
\(=2{ (-1) }^{ 3 }+{ (-1) }^{ 2 }-2(-1)-1\)
\(\\ =-2+1+2-1=0\)
\(\therefore\) By factor theorem, g(x) is a factor of p(x)
7.
\(p\left( -\frac { m }{ l } \right) =l\left( -\frac { m }{ l } \right) +m=-m+m=0\)
\(\therefore -\frac { m }{ l } \) is a zero of p(x)
8.
\(\therefore \ p(0)={ (0) }^{ 2 }-(0)+1=1\)
\( p(1)={ (1) }^{ 2 }-(1)+1=1\)
and \(p(2)={ (2) }^{ 2 }-(2)+1=4-2+1=3\)
9.
quadratic
10.
This expression is not a polynomial because in the term \(\frac { 2 }{ y } \) , the exponent of y is (- 1) which is not a whole number.
11.
x2-3x=y, then
y2-8y-20=y2-10y+2y-2
=y(y-10)+2(y-10)
=(y-10)(y+2)
=(x2-3x-10)(x2-3x+2)
=(x2-5x+2x-10)(x2-2x-x+2)
=[x(x-5)+2(x-5)][x(x-2)-1(x-2)]
=[(x-5)(x+2)][(x-1)(x-2)]
=(x-1)(x-2)(x+2)(x-5)
12.
We know that
\({ \left( x+\frac { 1 }{ x } \right) }^{ 3 }={ x }^{ 3 }+\frac { 1 }{ { x }^{ 3 } } +3(x)\left( \frac { 1 }{ x } \right) \left( x+\frac { 1 }{ x } \right) \)
\(\Rightarrow { \left( x+\frac { 1 }{ x } \right) }^{ 3 }={ x }^{ 3 }+\frac { 1 }{ { x }^{ 3 } } +3\left( x+\frac { 1 }{ x } \right) .\)
\(\Rightarrow { (7) }^{ 3 }={ x }^{ 3 }+\frac { 1 }{ { x }^{ 3 } } +3(7)\)
\(\Rightarrow { x }^{ 3 }+\frac { 1 }{ { x }^{ 3 } } =322\)
13.
We know that
\((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\)
\(\Rightarrow (a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)\)
\(\Rightarrow (a+b+c)^2=280+2\frac{9}{2}=289=(17)^2\)
\(\Rightarrow a+b+c=17\) Extracting square root of both sides
\(\Rightarrow (a+b+c)^3=(17)^3=4913\)
14.
\(p(z)={ 3z }^{ 2 }=4z+\sqrt { 17 } \)
\(\therefore p(3)=3{ (3) }^{ 2 }-4(3)+\sqrt { 17 }\)
\(=15+\sqrt { 17 }\)
15.
7
16.
\({ \left( \frac { 8 }{ 15 } \right) }^{ 3 }-{ \left( \frac { 1 }{ 3 } \right) }^{ 3 }-{ \left( \frac { 1 }{ 5 } \right) }^{ 3 }=\frac { x }{ 75 } \) .....(i)
Let \(\frac { 8 }{ 15 } =a,\frac { -1 }{ 3 } =b,\frac { -1 }{ 5 } =c\)
\(a+b+c=\frac { 8 }{ 15 } -\frac { 1 }{ 3 } -\frac { 1 }{ 5 } \)
\(=\frac { 8-5-3 }{ 15 } =0\)
\(\therefore\) a3+b3+c3=3abc .......(ii)
Using eqn. (i) and (ii), we get
\(3\times \frac { 8 }{ 15 } \times \frac { -1 }{ 3 } \times \frac { -1 }{ 5 } =\frac { x }{ 75 } \)
x=8.
17.
x4-y4=(x2)2-(y2)2
=(x2-y2)(x2+y2)
=(x-y)(x+y)(x2+y2).
18.
Let, x2+7=p and 2x-1=q, then given expression
=12p2-8pq-15q2
=12p2-18pq+10pq-15q2
=6p(2p-3q)+5q(2p-3q)
=(2p-3q)(6p+5q)
=[2(x2+7)-3(2x-1][6(x2+7)+5(2x-1)]
=(2x2+14-6x+3)(6x2+42+10x-5)
=(2x2-6x+17)(6x2+10x+37).
19.
(2x-3y+z)2=[2x+(-3y)+z]2
=(2x)2+(-3y)2+z2+2\(\times\) 2x\(\times\)(-3y)+2\(\times\)(-3y)\(\times\)z+2\(\times\)2x\(\times\)z
= 4x2+9y2+z2-12xy-6yz+4xz.
20.
p(x) = x3+x2+x+1
Put, \(x-\frac { 1 }{ 2 } =0\)
\(\Rightarrow \quad x=\frac { 1 }{ 2 } in\quad p(x)\)
Remainder = \(p\left( \frac { 1 }{ 2 } \right) \)
\(={ \left( \frac { 1 }{ 2 } \right) }^{ 3 }+{ \left( \frac { 1 }{ 2 } \right) }^{ 2 }+\left( \frac { 1 }{ 2 } \right) +1\)
\(=\frac { 1 }{ 8 } +\frac { 1 }{ 4 } +\frac { 1 }{ 2 } +1+1\)
\(=\frac { 1+2+4+8 }{ 8 } \)
\(=\frac { 15 }{ 8 } \)
21.
18
22.
1
23.
\(21a^2+\frac{1}{3}+8a\)
24.
(i) \((7a+5b)^2\)
(ii) \((\frac{5}{2}x+\frac{y}{3})(\frac{5}{2}x-\frac{y}{3})\)
25.
(x+1)(x+2)(x+10)
26.
\(-\frac{3}{2}\)
27.
8
28.
Quotient = 3x - 2
Remainder = 1
29.
binomial
30.
1
31.
\((x-\frac{1}{x})(x+\frac{1}{x})(x^2+\frac{1}{x^2})\)
\((x^2-\frac{1}{x^2})(x^2+\frac{1}{x^2})=x^4-\frac{1}{x^4}\)
32.
\(a^3+b^3+c^3-3abc\)
\(=(a+b+c)(z^2+b^2+c^2-ab-bc-ca)\)
\(=(a+b+c)(a+b+c)^2-3(ab+bc+ca)\)
\(=5(5)^2-3(10)\)
\(=-25\)
33.
\(42+y-y^2=42+7y-6y-y^2\)
\(=7(6+y)-y(6+y)\)
\(=(6+y)(7-y)\)
34.
If a+b+c=0 then \(a^3+b^3+c^3=3abc\)
35.
\(x+p=0\ \Rightarrow \ x=-p\)
By factor theorem,
\((-p)^2+p(-p)+3-p=0 \ \Rightarrow \ p=3\)
36.
\(x+1=0\ \ Rightarrow \ x=-1\)
\(\therefore\) Remainder=\((-1)^11+101=-1+101\)
=100
37.
Remainder theorem
38.
\(p(\frac{1}{3})=(\frac{1}{3})^2-(\frac{1}{3})+2=\frac{16}{9}\)
39.
\(f(x)=0 \quad \Rightarrow \quad x^2+2x+0\)
\(\Rightarrow \quad x(x+2)=0\quad \Rightarrow\quad x=0,-12\)
40.
Coefficient of \(x^2=(3)(4)+(-5)(4)=-8\)
41.
\((x-7)^3-x^3=-21x^2+147x-343\)
42.
Highest power of x=4
43.
\(y^2+2\) has two terms; three terms in (b) fractional power of y in (c) and (d);
44.
\(\frac{1}{y}=y^-1\) has negative exponent so, not a polynomial
45.
(a)
\(x^2+y^2\)
46.
( )
4x2+y2+4xy+8x+4y+4 = (2x)2+(y)2+(2)2+2\(\times\)2x\(\times\)y+2\(\times\)2x\(\times\)2+2\(\times\)y\(\times\)2
= (2x+y+2)2.
Factor is 2x + y + 2.
47.
( )
8y3-125x3=(2y)3-(5x)3
=(2y - 5x)(4y2+10xy+25x2)
48.
( )
20x2-9x+1 = 20x2-5x-4x+1
=5x(4x-1)-1(4x-1)
=(4x-1)(5x-1)
49.
( )
x2-3x=x(x-3)
50.
( )
Constant polynomial is 7.
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