Question

Pls answer this fast
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Answer 1

1) The constant polynomial

2)(i) Let us assume the division of 6x2 + 2x + 2 by 2

Here, p(x) = 6x2 + 2x + 2

g(x) = 2

q(x) = 3x2 + x + 1

r(x) = 0

Degree of p(x) and q(x) is same i.e. 2.

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

Or, 6x2 + 2x + 2 = 2x (3x2 + x + 1)

Hence, division algorithm is satisfied.

(ii) Let us assume the division of x3+ x by x2,

Here, p(x) = x3 + x

g(x) = x2

q(x) = x and r(x) = x

Clearly, the degree of q(x) and r(x) is the same i.e., 1.

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

x3 + x = (x2 ) × x + x

x3 + x = x3 + x

Thus, the division algorithm is satisfied.

(iii) Let us assume the division of x3+ 1 by x2.

Here, p(x) = x3 + 1

g(x) = x2

q(x) = x and r(x) = 1

Clearly, the degree of r(x) is 0.

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

x3 + 1 = (x2 ) × x + 1

x3 + 1 = x3 + 1

Thus, the division algorithm is satisfied.

3)real number

4)X - 1 is a factor of 4x^3 + 3x^2 -4x +k

then x=1 is one root of 4x^3 + 3x^2 -4x +k

put x= 1

4x^3 +3x^2 -4x +k = 0

=> 4 (1)^3 +3 (1)^2-4 (1) +k =0

=> 4 + 3 - 4 + k = 0

=> k = -3

5)Trinomial

6)The degree of the polynomial is zero. Because root 2can be written as root root 2x so x0, and the highest power of x of this polynomial is zero

7)Remainder is 50

8)it has no degree

9)3

10)Third degree polynomials are also known as cubic polynomials. Cubics have these characteristics: One to three roots. Two or zero extrema.