Question

Determine the value of k for which the polynomial 2x4 – kx3 + 4x2 + 2x + 1 is divisible by (1 – 2x).

Answer 1

$Given \: polynomial :\\p(x) =2x^{4}-kx^{3}+4x^{2}+2x+1\: and \: g(x) = (1-2x)$

$If \: p(x) \:divided \:by \: g(x) \:then \: p(\frac{1}{2}) = 0$

$p(\frac{1}{2}) = 0$

$\implies 2\Big(\frac{1}{2}\Big)^{4} - k\Big(\frac{1}{2}\Big)^{3}+4\Big(\frac{1}{2}\Big)^{2}+2\Big(\frac{1}{2}\Big)^{4}+1 = 0$

$\implies \frac{2}{16} - \frac{1}{8} k+ \frac{4}{4} + \frac{2}{2} + 1 = 0$

$\implies \frac{1}{8} - \frac{1}{8}k+ 1 + 1 + 1 =0$

$\implies -\frac{1}{8}k = \frac{1}{8} - 3$

$\implies -\frac{1}{8}k = \frac{1-24}{8}$

$\implies k = \frac{-23}{8} \times \frac{(-8)}{1}$

$\implies k = 23$

Therefore.,

$\red{ Value \:of \: k } \green {= 23 }$

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Answer 2

Answer:

Step-by-step explanation:

ANSWER:

As the polynomial  is divisible by (1 – 2x),

That also means

1-2x = 0

-2x = -1 __________( Minus and Minus gets cancelled)

x = 1/2

Placing the values of x in the polynomial,

$2x^4 - kx^3 + 4x^2 + 2x + 1 = 0$

$\implies 2(1/2)^4 - k(1/2)^3 + 4 (1/2)^2 +1+1 = 0$

$\implies\frac{2}{16} - \frac{k}{8} + \frac{4}{4} +2 = 0$

$\implies \frac{1-k}{8}+3 = 0$

1-k /8 = -3

1-k = -24

k = 23

k = 23 is the answer.

Things to Note:-

• Whenever this type of question comes always find the value of x first
• Then the second step is to place the value of x in the equation
• And as the equation is fully divisible with no remainder that is remainder is zero, the equation will be equal to zero.
• Then at last find the given constant.