Question

1. Determine which of the following polynomials has (x + 1) a factor
(i)x3 + x2 + x + 1
(ii) x4+ x3+ x2 + x + 1
(ii) x4 + 3x3+ 3x2 + x + 1
(iv) x - x - (2 + √2 ) x + √2​

Answer 1

Answer:

x^3+x^2+x+1 only has x+1 a factor

Step-by-step explanation:

Zero of x+1 is -1

i)let p(x)=x^3+x^2+x+1

p(-1)=(-1)^3+(-1)^2+(-1)+1

p(-1)= -1+1-1+1

p(-1)=0

ii)let p(x)=x^4+x^3+x^2+x+1

p(-1)=(-1)^4+(-1)^3+(-1)^2+(-1)+1

p(-1)=1-1+1-1+1

p(-1)=1

iii)let p(x)=x^4+3x^3+3x^2+x+1

p(-1)=(-1)^4+3(-1)^3+3(-1)^2+(-1)+1

p(-1)=1-3+3-1+1

p(-1)=1

iv)let p(x)=x-x-(2+root2)x+root2

(-1)=x-x-2x-root2x+root2

(-1)=(-1)-(-1)-2(-1)-root2(-1)+root2

(-1)= -1+1+2+root2+root2

(-1)=2+root4

Hope this is helpful for you

Plz mark it as brainlist answer

Answer 2

The zero of x + 1 is -1.

(i) Let p (x) = x3 + x2 + x + 1

∴ p (-1) = (-1)3 + (-1)2 + (-1) + 1 .

= -1 + 1 – 1 + 1

⇒ p (- 1) = 0

So, (x+ 1) is a factor of x3 + x2 + x + 1.

(ii) Let p (x) = x4 + x3 + x2 + x + 1

∴ P(-1) = (-1)4 + (-1)3 + (-1)2 + (-1)+1

= 1 – 1 + 1 – 1 + 1

⇒ P (-1) ≠ 1

So, (x + 1) is not a factor of x4 + x3 + x2 + x+ 1.

(iii) Let p (x) = x4 + 3x3 + 3x2 + x + 1 .

∴ p (-1)= (-1)4 + 3 (-1)3 + 3 (-1)2 + (- 1) + 1

= 1 – 3 + 3 – 1 + 1 = 1

⇒ p (-1) ≠ 0

So, (x + 1) is not a factor of x4 + 3x3 + 3x2 + x+ 1.

(iv) Let p (x) = x3 – x2 – (2 + √2) x + √2

∴ p (- 1) =(- 1)3- (-1)2 – (2 + √2)(-1) + √2

= -1 – 1 + 2 + √2 + √2

= 2√2

⇒ p (-1) ≠ 0

So, (x + 1) is not a factor of x3 – x2 – (2 + √2) x + √2.