Free Polynomials 02 Practice Test - 9th Grade
If xg+1xa is a polynomial, what could be the value of a?
Solution : C
For an expression to be a polynomial, the exponents of x should be whole numbers. The above expression can be written as xg+x−a . For this to be a polynomial, −a should be a whole number. This is possible if a is a negative integer or zero.
If ax2+bx+c is a monomial in x , what can be said about a, b and c?
Solution : C
Since a monomial can contain only one term, only one of a, b and c can be non-zero to leave a single term.
If axn is a zero degree polynomial in x, what can be said about a?
Solution : D
A zero degree polynomial is essentially a constant. i.e., any non-zero real number.
So, if axn is a zero degree polynomial, then
axn = ax0 = a
So, n has to be zero, and a should not be zero.
Find the factors of x3+2x2+2x+1
Solution : A and B
Let p(x)=(x3+2x2+2x+1) be the given polynomial
By observation, we can see that by putting
∴x=−1 is the root of p(x)
⇒(x+1) is a factor of p(x)
To find the other factor, we perform long division of p(x) by (x+1)
∴ The two factors of p(x) are (x+1) and (x2+x+1)
Given the area of rectangle is A=25a2−35a+69. The length is given as (5a−3).Find the width of the rectangle.
Solution : B
Let y be the width.
Given: Area =25a2−35a+69
We know that, area of a rectangle = Length ×breadth
By long division method,
5a−35a−23)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯25a2−35a+69 25a2−12a − + ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ −23a+69 −23a+69 + − ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 0
∴y=5a−23Width of the rectangle =5a−23
If a polynomial(x2+ax−c)cuts x-axis at 2 and -4, the value of a+c is
Suppose f(x) is the polynomial.
Now since it cuts x-axis at 2 and -4, the roots of f(x) are 2 and -4.
Sum of roots = −a=2−4=−2
Product of roots = −c=2×−4=−8
Which of the following are the factors of 1012−992
Solution : B and C
Using the algebraic Identity (a2−b2)=(a−b)(a+b)
⇒(a−b) and (a+b) are factors of a2−b2.
We find that 2 and 200 both are the factors of (1012−992).
Factor Theorem is equivalent to Remainder Theorem when remainder is
Solution : C
If f(x) is a polynomial and is divided by (x-a), then if f(a) = k, this k will be the remainder, as stated by the Remainder Theorem. If the value of k is 0, then (x-a) is the factor of f(x), as indicated by the Factor Theorem. Hence, factor theorem is equivalent to remainder theorem when remainder is 0.
Select the values which the coefficients and exponents of a polynomial can take? (Options are given by comma separating them in the form - (coefficient,exponent))
Solution : A and C
Any real number can be a coefficient in a polynomial. Thus, rational and irrational numbers are valid coefficients.
Exponents in polynomials can only be whole numbers.
If f(x) = (x - a)(x - b) , then if a and b are the zeros of the polynomial f(x), f(a) = f(b).
Solution : A
Since a and b are the zeroes of the polynomial f(x), both f(a) = 0 and f(b) = 0, i.e. a and b will be the roots of the equation f(x) = 0.
Thus, f(a) = f(b) = 0. Hence, the given statement is true.