Free Polynomials 03 Practice Test - 9th Grade
Question 1
Which of the following given sequences can be represented as a polynomial in one variable?
1, 2, 3,...
1, 3, 5...
2, 4, 6...
1, 4, 9...
SOLUTION
Solution : A, B, C, and D
1, 2, 3... sequence can be represented using the polynomial x.
1, 3, 5... sequence can be represented using the polynomial 2x+1.
2, 4, 6... sequence can be represented using the polynomial 2x.
1, 4, 9... sequence can be represented using the polynomial x2.
All of these are polynomials in one variable i.e., x.
Question 2
If x+1x=2, find the value of x64+1x64
32
16
8
2
SOLUTION
Solution : D
x+1x=2
Squaring both sides, we get x2+1x2+2.x.1x=4
So, x2+1x2=4−2=2
Again Squaring both sides, we get x4+1x4+2.x2.1x2=4
⇒x4+1x4=4−2=2
Similarly, x8+1x8=x32+1x32=x64+1x64=2
Question 3
If (x2+y2+z2)=(xy+yz+zx), what is the value of x3+y3+z3?
3xyz
−3xyz
2xyz
−2xyz
SOLUTION
Solution : A
We know that
x3+y3+z3−3xyz=(x+y+z)(x2+y2+z2−xy−yz−zx)−−−(1)
According to question, (x2+y2+z2)=(xy+yz+zx).
∴ RHS will be zero in equation (1)
So,x3+y3+z3=3xyz
Question 4
If (x−1) is a factor of ax3−bx2+cx, what is the value of a3−b3+c3?
2abc
-2abc
3abc
-3abc
SOLUTION
Solution : D
p(x)=ax3−bx2+cx
If (x - 1) is a factor then x = 1 is the zero of polynomial
p(1) = 0p(1)=a−b+c=0
It can be written as a+(−b)+c=0
We know that if a+b+c=0, then a3+b3+c3=3abc
So, a3+(−b)3+c3=3a(−b)c=−3abc
Question 5
If (x2−y2)=18 and (x−y)=3, find the value of 16x2y2.
SOLUTION
Solution : D
(x2−y2)=(x−y)(x+y)=18
(x−y)=3−−−−(i)
(x+y)=183=6−−−−(ii)Adding equation (i) and (ii) we get
2x=9
x=92
Substituting the value of x in equation (ii) we get
y=32
16x2y2=16×(92)2×(32)2=729
Question 6
In a2x2+ax, if 'a' is constant and 'x' is a variable, this expression would become a polynomial in
SOLUTION
Solution :In this expression, a2x2 and ax are the two terms. Since it is given that 'a' is a constant and 'x' is a variable, we can see that there is only one variable 'x' in the polynomial thus making it a polynomial in one variable.
Question 7
If xg+1xa is a polynomial in one variable and if it is given that g = a, for what value of 'a' is this a polynomial?
SOLUTION
Solution :xg+1xa=xg+x−a
Since g = a,
xg+1xa=xa+x−a
For the expression xa+x−a to be a polynomial, a and -a must be whole number so that the expression 1xa = 1 , hence a should be equal to 0 .
Thus, a = 0.
When a=0, the given polynomial will get simplified further as:
xg+1xa=xa+x−a
⇒ xg+1xa=x0+x−0
⇒ xg+1xa=x0+x0
⇒ xg+1xa=1+1=2
'2' is a constant polynomial in one variable with degree zero . Because 2=2x0.
So, 'a' must be equal to zero so that given expression is a polynomial in variable.
Question 8
Factorize : x2−7x+12
(x-3)(x-4)
SOLUTION
Solution : A
We have to break up 7x in the given expression into two parts in such a way that their sum is 7x and the product of their coefficients is 12 .
x2−7x+12
=x2−4x−3x+12
=x(x−4)−3(x−4)
=(x−3)(x−4)
Thus x2−7x+12 can be factorised as (x−3)(x−4)
Question 9
p(x)=x3+8x2−7x+12 and g(x)=x−1. If p(x) is divided by g(x), it gives q(x) and r(x) as quotient and remainder respectively. If a is the degree of q(x) and b is the degree of r(x), (a−b)=?.
1
SOLUTION
Solution : B
Remainder =r(x)=p(1)=13+8×12−7×1+12=14
The degree of a polynomial is the highest degree of its terms when the polynomial is expressed in its canonical form consisting of a linear combination of monomials.
So, Degree of r(x)=b=0
q(x)=p(x)g(x)
Now , p(x)=x3+8x2−7x+12
g(x)=(x−1)
Performing long division
x2+9x+2x−1)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x3+8x2−7x+12 x3−x2 (−) (+) ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 9x2−7x 9x2−9x (−) (+) ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ +2x+12 2x−2 (−) (+) ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 14
∴q(x)=p(x)g(x)=x2+9x+2
So, degree of q(x)=a=2
Hence, a−b=2−0=2
Question 10
The value of (2x−3y)3+(x+3y)3+3(2x−3y)2(x+3y)+3(2x−3y)(x+3y)2=
SOLUTION
Solution : A
Substitute 2x−3y=a and (x+3y)=b
(2x−3y)3+(x+3y)3+3(2x−3y)2(x+3y)+3(2x−3y)(x+3y)2
=a3+b3+3a2b+3ab2
=(a+b)3
=(2x−3y+x+3y)3
=(3x)3
=27x3