Question

Check whether (x - 2) is a factor of x2 - 2x7 - 5x + 4​

Answer 1

GIVEN :-

• p(x) = x² - 2x⁷ - 5x + 4
• g(x) = x - 2

TO CHECK :-

• whether g(x) is a factor of p(x).

SOLUTION :-

✝ ➠ Factor theorem :- x - a is a factor of the polynomial p(x) , If p(a) = 0. Also, If x - a is a Factor of p(x) , Then p(a) = 0.

◉ Let x - 2 = 0

◉ x = 2

➫ p(x) = x² - 2x⁷ - 5x + 4

➫ p(2) = (2)² - 2 × (2)⁷ - 5 × 2 + 4

➫ 4 - 2 × 128 - 10 + 4

➫ 4 - 256 - 10 + 4

➫ 8 - 266

➫ -258

◉ Since the Remainder is -258

☛ Hence g(x) is not Factor of p(x)

ADDITIONAL INFORMATION :-

➠ Remainder theorem :- If p(x) is is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial x - a , Then the reminder is p(a).

Answer 2

ANSWER✔

$\large\underline\bold{GIVEN,}$

$\sf\dashrightarrow p(x)=x^2-2x^7-5x+4$

$\sf\dashrightarrow p(x)= -2x^7+x^2-5x+4$

$\sf\dashrightarrow g(x)=(x-2)$

$\large\underline\bold{TO\:FIND,}$

$\sf\dashrightarrow WHETHER\:G(X)\:IS \:THE\:FACTOR\:OF\:P(X)$

$\large\underline\bold{SOLUTION,}$

TAKING G(X)=(X-2)

$\sf\implies x-2=0$

$\sf\implies x=0+2$

$\sf\implies x=2$

$\large{\boxed{\bf{ x=2}}}$

NOW,

SUBSTITUTING THE VALUE OF X IN P(X),

$\sf\therefore p(x)=x^2-2x^7-5x+4$

$\sf\therefore x=2$

$\sf\therefore p(2)= -2(2)^7+(2)^2-5(2)+4$

$\sf\implies (-2)(128)+(4)-(10)+4=0$

$\sf\implies (-256)-6+4=0$

$\sf\implies -256-2=0$

$\sf\implies -258=0$

$\sf\therefore L.H.S \neq R.H.S$

$\sf\large\bold\therefore g(x)\: is \:not \:factor \:of\:p(x)$

✯HENCE, CHECKED✔

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