Question

x + 3 is a factor of x⁴ + 4x³ - 7x² - 22 x +m check whether x + 4 is also a factor of this polynomial​

Answer 1

Step-by-step explanation:

Given, polynomial is f(x) = x⁴ + 4x³ - 7x² - 22x + m

Also given, x + 3 is a factor of f(x)

=> f(-3) = 0

=> (-3)⁴ + 4(-3)³ - 7(-3)² - 22(-3) + m = 0

=> 81 + 4(-27) - 7(9) + 66 + m = 0

=> 81 - 108 - 63 + 66 + m = 0

=> -24 + m = 0

=> m = 24

f(-4) = (-4)⁴ + 4(-4)³ - 7(-4)² - 22(-4) + 24

=> 256 + 4(-64) - 7(16) + 88 + 24

=> 256 - 256 - 112 + 112 = 0

=> f(-4) = 0

=> x + 4 is also a factor of the given polynomial

Answer 2

ANSWER✔

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

$\dashrightarrow P(x)= x^4+4x^3-7x^2-22+m$

$\dashrightarrow g(x)= x+4$

$\dashrightarrow another\:g(x)= x+3\: which\:is\:factor\:of\:p(x)$

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

$\dashrightarrow x+4\:is\:a\:factor\:of\:x^4+4x^3-7x^2-22+m$

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

$\therefore finding\:the\:value\:of\:m,$

$\dashrightarrow x+3=0 \\ \implies x=-3$

$\implies x^4+4x^3-7x^2-22+m=0$

$\implies (-3)^4-4(-3)^3-7(-3)^2-22(-3)+m=0$

$\implies 81-108-63+66+m=0$

$\implies -27+3+m=0$

$\implies -24+m=0$

$\implies m=24$

$\large{\boxed{\bf{ \star\:\: m=24\:\: \star}}}$

now,

FINDING WHETHER x-4 IS THE FACTOR OF THE GIVEN

POLYNOMIAL OR NOT.

$\dashrightarrow x+4=0\\ \implies x=-4$

$\implies x^4+4x^3-7x^2-22x+24$

$\implies (-4)^4+4(-4)^3-7(-4)^2-22(-4)+24$

$\implies 256+4(-64)-7(16)+88+24$

$\implies 256-256-112+112$

$\implies 0+0$

$\therefore remainder= 0\:--\boxed{x+4\:is\:a\:factor\:of\: x^4+4x^3-7x^2-22x+24}$

$\large\underline\bold{x+4\:IS\:ALSO\:FACTOR\:OF\:x^4+4x^3-7x^2-22x+24}$

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