Math Polynomials

Question 1

The zeroes of x2–2x –8 are
(a) (2,-4)

(b) (4,-2)

(c) (-2,-2)

(d) (-4,-4)

Answer 1

(b) (4,-2)
Explanation: x2–2x –8 = x2–4x + 2x –8

= x(x–4)+2(x–4)

= (x-4)(x+2)

Therefore, x = 4, -2.

Question 2

What is the quadratic polynomial whose sum and the product of zeroes is √2, ⅓ respectively?

(a) 3x2-3√2x+1

(b) 3x2+3√2x+1

(c) 3x2+3√2x-1

(d) None of the above

Answer 2

(a) 3x2-3√2x+1

Explanation: Sum of zeroes = α + β =√2

Product of zeroes = α β = 1/3

∴ If α and β are zeroes of any quadratic polynomial, then the polynomial is;

x2–(α+β)x +αβ

= x2 –(√2)x + (1/3)

= 3x2-3√2x+1

Question 3

If the zeroes of the quadratic polynomial ax2+bx+c, c≠0 are equal, then

(a) c and b have opposite signs

(b) c and a have opposite signs

(c) c and b have same signs

(d) c and a have same signs

Answer 3

(d) c and a have same signs

Explanation:

For equal roots, discriminant will be equal to zero.

b2 -4ac = 0

b2 = 4ac

ac = b2/4

ac>0 (as square of any number cannot be negative)

Question 4

The degree of the polynomial, x4 – x2 +2 is

(a) 2

(b) 4

(c) 1

(d) 0

Answer 4

(b) 4
Explanation: Degree is the highest power of the variable in any polynomial.

Question 5

If one of the zeroes of cubic polynomial is x3+ax2+bx+c is -1, then product of other two zeroes is:

(a) b-a-1

(b) b-a+1

(c) a-b+1

(d) a-b-1

Answer 5

(b) b-a+1
Explanation: Since one zero is -1, hence;

P(x) = x3+ax2+bx+c

P(-1) = (-1)3+a(-1)2+b(-1)+c

0 = -1+a-b+c

c=1-a+b

Product of zeroes, αβγ = -constant term/coefficient of x3

(-1)βγ = -c/1

c=βγ

βγ = b-a+1

Question 6

If p(x) is a polynomial of degree one and p(a) = 0, then a is said to be:

(a) Zero of p(x)

(b) Value of p(x)

(c) Constant of p(x)

(d) None of the above

Answer 6

(a) Zero of p(x)

Explanation: Let p(x) = mx+n

Put x = a

p(a)=ma+n=0

So, a is zero of p(x).

Question 7

Zeroes of a polynomial can be expressed graphically. Number of zeroes of polynomial is equal to number of points where the graph of polynomial is:

(a) Intersects x-axis

(b) Intersects y-axis

(c) Intersects y-axis or x-axis

(d) None of the above

Answer 7

(a) Intersects x-axis

Question 8

A polynomial of degree n has:

(a) Only one zero

(b) At least n zeroes

(c) More than n zeroes

(d) At most n zeroes

Answer 8

(d) At most n zeroes

Explanation: Maximum number of zeroes of a polynomial = Degree of the polynomial

Question 9

The number of polynomials having zeroes as -2 and 5 is:

(a) 1

(b) 2

(c) 3

(d) More than 3

Answer 9

(d) More than 3

Explanation: The polynomials x2-3x-10, 2x2-6x-20, (1/2)x2-(3/2)x-5, 3x2-9x-30, have zeroes as -2 and 5.

Question 10

Zeroes of p(x) = x2-27 are:

(a) ±9√3

(b) ±3√3

(c) ±7√3

(d) None of the above

Answer 10

(b) ±3√3

Explanation: x2-27 = 0

x2=27

x=√27

x=±3√3

Question 11

Given that two of the zeroes of the cubic polynomial ax3 + bx2 + cx + d are 0, the third zero is

(a) -b/a

(b) b/a

(c) c/a

(d) -d/a

Answer 11

Answer: (a) -b/a

Explanation:

Let α be the third zero.

Given that two zeroes of the cubic polynomial are 0.

Sum of the zeroes = α + 0 + 0 = -b/a

α = -b/a

Question 12

If one zero of the quadratic polynomial x2 + 3x + k is 2, then the value of k is

(a) 10

(b) –10

(c) 5

(d) –5

Answer 12

Answer: (b) -10

Explanation:

Given that 2 is the zero of the quadratic polynomial x2 + 3x + k.

⇒ (2)2 + 3(2) + k = 0

⇒ 4 + 6 + k = 0

⇒ k = -10

Question 13

A quadratic polynomial, whose zeroes are –3 and 4, is

(a) x² – x + 12

(b) x² + x + 12

(c) (x²/2) – (x/2) – 6

(d) 2x² + 2x – 24

Answer 13

Answer: (c) (x²/2) – (x/2) – 6

Explanation:

Let the given zeroes be α = -3 and β = 4.

Sum of zeroes, α + β= -3 + 4 = 1

Product of Zeroes, αβ = -3 × 4 = -12

Therefore, the quadratic polynomial = x² – (sum of zeroes)x + (product of zeroes)

= x² – (α + β)x + (αβ)

= x² – (1)x + (-12)

= x² – x – 12

Dividing by 2,

= (x²/2) – (x/2) – 6

Question 14

The zeroes of the quadratic polynomial x2 + 99x + 127 are

(a) both positive

(b) both negative

(c) one positive and one negative

(d) both equal

Answer 14

Answer: (b) both negative

Explanation:

Given quadratic polynomial is x2 + 99x + 127.

By comparing with the standard form, we get;

a = 1, b = 99 and c = 127

a > 0, b > 0 and c > 0

We know that in any quadratic polynomial, if all the coefficients have the same sign, then the zeroes of that polynomial will be negative.

Therefore, the zeroes of the given quadratic polynomial are negative.

Question 15

The zeroes of the quadratic polynomial x2 + 7x + 10 are

(a) -4, -3

(b) 2, 5

(c) -2, -5

(d) -2, 5

Answer 15

Answer: (c) -2, -5

Explanation:

x2 + 7x + 10 = x2 + 2x + 5x + 10

= x(x + 2) + 5(x + 2)

= (x + 2)(x + 5)

Therefore, -2 and -5 are the zeroes of the given polynomial.

Question 16

If the discriminant of a quadratic polynomial, D > 0, then the polynomial has

(a) two real and equal roots

(b) two real and unequal roots

(c) imaginary roots

(d) no roots

Answer 16

(b) two real and unequal roots
If the discriminant of a quadratic polynomial, D > 0, then the polynomial has two real and unequal roots.

Question 17

If on division of a polynomial p(x) by a polynomial g(x), the quotient is zero, then the relation between the degrees of p(x) and g(x) is

(a) degree of p(x) < degree of g(x)

(b) degree of p(x) = degree of g(x)

(c) degree of p(x) > degree of g(x)

(d) nothing can be said about degrees of p(x) and g(x)

Answer 17

(a) degree of p(x) < degree of g(x)
Explanation:

We know that, p(x)= g(x) × q(x) + r(x)

Given that, q(x) = 0

When q(x) = 0, then r(x) = 0

So, now when we divide p(x) by g(x),

Then p(x) should be equal to zero.

If r(x) = 0, then the degree of p(x) < degree of g(x).

Question 18

By division algorithm of polynomials, p(x) =

(a) g(x) × q(x) + r(x)

(b) g(x) × q(x) – r(x)

(c) g(x) × q(x) × r(x)

(d) g(x) + q(x) + r(x)

Answer 18

(a) g(x) × q(x) + r(x)

By division algorithm of polynomials, p(x) = g(x) × q(x) + r(x).

Question 19

The product of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is

(a) -b/a

(b) c/a

(c) -d/a

(d) -c/a

Answer 19

(c) -d/a

The product of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is -d/a.

Question 20

If the graph of a polynomial intersects the x-axis at three points, then it contains ____ zeroes.

(a) Three

(b) Two

(c) Four

(d) More than three

Answer 20

(a) Three

If the graph of a polynomial intersects the x-axis at three points, then it contains three zeroes.