Question

2.
If P = 2x4x6x...X20 and Q = 1x3x5x...X19, then the HCF of P and Q is
3³x5²
3⁴x5²×7
3⁴x5
3³x5x7​

Answer 1

Answer:

Step 1: Recall the Factor theorem.

Factor theorem:

f(x) is a polynomial of degree  n≥1  and 'a' is some real number, If f(a) = 0, then (x - a) is a factor of f(x).

Step 2: Use the factor theorem to simplify

Given that

 f(x)=2x3+px2+qx−14

(x - 1) and (x + 2) are factors of f(x)

According to the factor theorem if (x - a) is a factor of f(x), then f(a) = 0

(x - a) = (x - 1)

a = 1

So,  f(a)=2x3+px2+qx−14=0 

 f(1)=2(1)3+p(1)2+q∗(1)−14=0

 f(1)=2+p+q−14=0

 f(1)=p+q−12=0 ...................(1)

(x - a) = (x + 2) = (x - (-2))

a = -2

 f(a)=2x3+px2+qx−14=0

 f(−2)=2(−2)3+p∗(−2)2+q∗(−2)−14=0

 f(−2)=2∗(−8)+p∗4−q∗2−14=0

 f(−2)=−16+4p−2q−14=0 

 f(−2)=4p−2q−30=0

f(- 2) = 2(2p - q - 15) = 0

f(- 2) = 2p - q - 15 = 0.....................(2)

Step 3: Solve the equations in two variables

p + q - 12 = 0 ........................(1)

2p - q - 15 = 0 ........................(2)

Add equation (2) and equation (1)

p + q - 12 = 0 ........................(1)

+ 2p - q - 15 = 0 ...................(2)

_____________________________________

3p + 0 - 27 = 0

3p = 27

p=273=9

Substitute the value p = 9 in equation (1)

p + q - 12 = 0 .

9 + q - 12 = 0

 q−3=0 

 q=3

Hence, value of p = 9 and q = 3

Step-by-step explanation:

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