Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

(i) 3x²-4x+15

(ii) y²+2

(iii) 3+x

(iv) x-

(v) x¹²+y³+t⁵⁰

Write the coefficient of x² in each of the following:

(i) 17 – 2x + 7x²

(ii) 9-12x+x³

(iii) x² – 3x + 4

(iv) x-7

Write the degrees of each of the following polynomials:

(i) 7x³+4x²-3x+12

(ii) 12 -x+2x³

(iii) 5y-

(iv) 7

(v) 0

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

(i) x+x²+7y² (ii) 3x-2

(iii) 2x+x²

(iv) 3y (v) t²+1

(vi) 7t⁴+4t³+3t-2

Classify the following polynomials as polynomials in one-variable, two variable etc:

(i) x²-xy+7y² (ii) x²-2tx+7y²-x+t

(iii) t³-3t²+4t-5 (iv) xy + yz + zx

Identify polynomials in the following:

(i) f(x) = 4x³-x²-3x+7

(ii) g(x) = 2x³-3x²+-1

(iii) p(x) = x²-x+9

(iv) q(x) = 2x²-3x++2

(v) h(x) = x⁴-+x-1

(vi) f(x) = 2++4x

Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

(i) f(x) =0 (ii) g(x) = 2x³-7x+4

(iii) h(x) = -3x+

(iv) p(x) = 2x²-x+4

(v) q(x) = 4x+3 (vi) r(x) = 3x³+4x²+5x-7

Give one example each of a binomial of degree 35, and of a monomial of degree 100

If f(x) = 2x³-13x²+17x+12, find

(i) f(2) (ii) f(-3) (iii) f(0)

Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases:

(i) f(x) = 3x+1; x =

(ii) f(x) = x²-1; x = 1, -1

(iii) g(x) = 3x²-2; x = , -

(iv) p(x) = x³-6x²+11x-6, x = 1,2,3

(v) f(x) = 5x-π, x=

(vi) f(x) = x², x=0

(vii) f(x) =1x+m, x=

(viii) f(x) = 2x+1, x =

If x = 2 is a root of the polynomial f (x)= 2x²-3x+7a, find the value of a.

If x = -1/2 is a zero of the polynomial p (x) = 8x³ - ax² - x + 2, find the value of a.

If x = 0 and x = -1 are the roots of the polynomial f (x) = 2x³-3x²+ax+b, find the value of a and b.

Find the integral roots of the polynomial f(x) = x³+6x²+11x+6.

Find rational roots of the polynomial f(x) = 2x³+x²-7x-6.

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):

f(x) = x³+4x²-3x+10, g(x) = x+4

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):

f(x) = 4x⁴-3x³-2x²+x-7, g(x) = x-1

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):

f(x) = 2x⁴-6x³+2x²-x+2, g(x) = x+2

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):

f(x) = 4x³-12x²+14x-3, g(x) = 2x-1

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):

f(x) = x³-6x²+2x-4, g(x) = 1-2x

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):

f(x) = x⁴-3x²+4, g(x) = x-2

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):

f(x) = 9x³-3x²+x-5, g(x) = x=

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):

f(x) = 3x⁴+2x³, g(x) = x+

If the polynomials 2x³+ax²+3x-5 and x³+x²-4x+a leave the same remainder when divided by x-2, find the value of a.

If the polynomials ax³+3x²-3x and 2x³-5x+a when divided by (x-4) leave the remainder R₁ and R₂ respectively. Find the value of a in each of the following cases, if

(i) R₁ = R₂ (ii) R₁ + R₂=0

(iii) 2R₁-R₂ = 0.

If the polynomials ax³+3x²-13 and 2x³-5x+a when divided by (x-2) leave the same remainder, find the value of a.

Find the remainder when x³+3x²+3x+1 is divided by

In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:

f(x) = x³-6x²+11x-6, g(x) = x-3

In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:

f(x) = 3x⁴+17x³+9x²-7x-10, g(x) = x+5

In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:

f(x) = x⁵+3x⁴-x³-3x²+5x+15, g(x) = x+3

In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:

f(x) = x³-6x²-19x+84, g(x) = x-7

In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:

f(x) = 3x³+x²-20x+12, g(x) = 3x-2

In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:

f(x) = 2x³-9x²+x+12, g(x) = 3-2x

In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:

f(x) = x³-6x²+11x-6, g(x) = x²-3x+2

Show that (x-2), (x+3) and (x-4) are factors of x³-3x²-10x+24.

Show that (x+4),(x-3) and (x-7) are factors of x³-6x²-19x+84.

For what value of a is (x-5) a factor of x³-3x²+ax-10.

Find the value of a such that (x-4) is a factor of 5x³-7x²-ax-28.

Find the value of a, if x+2 is a factor of 4x⁴+2x³-3x²+8x+5a.

Find the value of k if x-3 is a factor of k²x³-kx²+3kx-k.

Find the value is of a and b, if x²-4 is a factor of ax⁴+2x³-3x²+bx-4.

Find α and β if x+1 and x+2 are factors of x³+3x²-2αx+β.

Find the value of p and q so that x⁴+px³+2x²-3x+q is divisible by (x²-1).

Find the value is of a and b, so that (x+1) and (x-1) are factors of x⁴+ax³-3x²+2x+b.

If x³+ax²-bx+10 is divisible by x²-3x+2, find the values of a and b.

If both x+1 and x-1 are factors of ax³+x²-2x+b, find the value of a and b.

What must be added to x³ - 3x² - 12x + 19 so that the result is exactly divisibly by x² + x - 6?

What must be subtracted from x³ - 6x² - 15x + 80, so that the result is exactly divisible by x² + x - 12?

What must be added to 3x³ + x² - 22x + 9 so that the result is exactly divisible by 3x² + 7x - 6?

If x-2 is a factor of each of the following two polynomials, find the values of a in each case.

(i) x³-2ax²+ax-1

(ii) x⁵-3x⁴-ax³+3ax²+2ax+4

In each of the following two polynomials, find the value of a, if x-a is a factor:

(i) x⁶-ax⁵+x⁴-ax³+3x-a+2.

(ii) x⁵-a²x³+2x+a+1.

In each of the following two polynomials, find the value of a, if x+a is a factor:

(i) x³+ax²-2x+a+4

(ii) x⁴-a²x²+3x-a

Using factor theorem, factorize each of the following polynomial:

x³+6x²+11x+6

Using factor theorem, factorize each of the following polynomial:

x³+2x²-x-2

Using factor theorem, factorize each of the following polynomial:

x³-6x²+3x+10

Using factor theorem, factorize each of the following polynomial:

x⁴-7x³+9x²+7x-10

Using factor theorem, factorize each of the following polynomial:

x⁴-2x³-7x²+8x+12

Using factor theorem, factorize each of the following polynomial:

x⁴+10x³+35x²+50x +24

Using factor theorem, factorize each of the following polynomial:

2x⁴-7x³-13x²+63x-45

Using factor theorem, factorize each of the following polynomial:

3x³-x²-3x+1

Using factor theorem, factorize each of the following polynomial:

x³-23x²+142x-120

Using factor theorem, factorize each of the following polynomial:

y³-7y+ 6

Using factor theorem, factorize each of the following polynomial:

x³-10x²-53x-42

Using factor theorem, factorize each of the following polynomial:

y³-2y²-29y-42

Using factor theorem, factorize each of the following polynomial:

2y³-5y²-19y+42

Using factor theorem, factorize each of the following polynomial:

x³+13x²+32x+20

Using factor theorem, factorize each of the following polynomial:

x³-3x²-9x-5

Using factor theorem, factorize each of the following polynomial:

2y³+y²-2y-1

Using factor theorem, factorize each of the following polynomial:

x³-2x²-x+2

Factorize each of the following polynomials:

(i) x³+13x²+31x-45 given that x+9 is a factor

(ii) 4x³+20x²+33x+18 given that 2x+3 is a factor.

Define zero or root of a polynomial.

If x = is a zero of the polynomial f(x) =8x³+ax²-4x+2, find the value of a.

Write the remainder when the polynomial f(x) =x³+x²-3x+2 is divided by x+1.

Find the remainder when x³+4x²+4x-3 is divided by x.

If x+1 is a factor of x³+a, then write the value of a.

If f(x) =x⁴-2x³+3x²-ax-b when divided by x-1, the remainder is 6, then find the value of a + b

If x-2 is factor of x²-3ax-2a, then a =

If x³+6x²+4x+k is exactly divisible by x+2, then k =

If x-a is a factor of x³-3x²a+2a²x+b, then the value of b is

If x¹⁴⁰+2x¹⁵¹ +k is divisible by x+1, then the value of k is

If x+2and x-1 are the factors of x³+10x²+mx+n, then the value of m and n are respectively

Let f(x) be a polynomial such that f = 0, then a factor of f(x) is

When x³-2x²+ax=b is divided by x²-2x-3, the remainder is x-6. The value of a and b respectively

One factor of x⁴+x²-20 is x²+5. The other factor is

If (x-1) is a factor of polynomial f(x) but not of g(x), then it must be a factor of

(x+1) is a factor of xⁿ+1 only if

If x+2 is a factor of x²+mx+14, then m =

If x -3 is a factor of x²-ax-15, then a =

If x²+x+1 is a factor of the polynomial 3x²+8x²+8x+3+5k, then the value of k is

If (3x-1)⁷ =a₇x⁷+a₆x⁶+a₅x⁵+….a₁x+a₀, then a₇+a₆+a₅+….+a₁+a₀=

If x⁵¹+51 is divide by x+1, the remainder is

If x+1 is a factor if the polynomial 2x²+kx, then k =

If x + a is a factor of x⁴-a²x²+3x-6a, then a =

The value of k for which x-1 is a factor of 4x³+3x²-4x+k, is

If both x-2 and x are factors of px²+5x+r, then

If x²-1 is a factor of ax⁴+bx³+cx²+dx+e, then