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Determine which of the following polynomials has (x+1) a facto

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Determine which of the following polynomials has $(x + 1)$ a factor:
(i) $x^{3} + x^{2} + x + 1$
(ii) $x^{4} + x^{3} + x^{2} + x + 1$
(iii) $x^{4} + 3 x^{3} + 3 x^{2} + x + 1$
(iv) $x^{3} - x^{2} - (2 + 2) x + 2$

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Determine which of the following polynomials has

(x + 1)

a factor:
(i)

x^{3} + x^{2} + x + 1

(ii)

x^{4} + x^{3} + x^{2} + x + 1

(iii)

x^{4} + 3 x^{3} + 3 x^{2} + x + 1

(iv)

x^{3} - x^{2} - (2 + 2) x + 2

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Text solutionVerified

Apply remainder theorem

x + 1 = 0

x = - 1

Put the value of

x = - 1

in all equations.

(i)

x^{3} + x^{2} + x + 1 = (- 1)^{3} + (- 1)^{2} + (- 1) + 1 = - 1 + 1 - 1 + 1 = 0

Then

x + 1

is the factor of equation

(ii)

x^{4} + x^{3} + x^{2} + x + 1 = (- 1)^{4} + (- 1)^{3} + (- 1)^{2} + (- 1) + 1 = 1 - 1 + 1 - 1 + 1 = 1

This is not zero.Then

x + 1

is not the factor of equation

(iii)

x^{4} + 3 x^{3} + 3 x^{2} + x + 1 = (- 1)^{4} + 3 (- 1)^{3} + 3 (- 1)^{2} + (- 1) + 1 = 1

This is not zero.Then

x + 1

is not the factor of equation

(iv)

x^{3} - x^{2} - (2 + 2) x + 2 = (- 1)^{3} - (- 1)^{2} - (2 + 2) (- 1) + 2 = - 1 - 1 + 2 - 2 + 2 = 0

This is zero. Then

x + 1

is the factor of equation

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Question Text	Determine which of the following polynomials has $(x + 1)$ a factor: (i) $x^{3} + x^{2} + x + 1$ (ii) $x^{4} + x^{3} + x^{2} + x + 1$ (iii) $x^{4} + 3 x^{3} + 3 x^{2} + x + 1$ (iv) $x^{3} - x^{2} - (2 + 2) x + 2$
Answer Type	Text solution:1
Upvotes	150

Determine which of the following polynomials has (x+1) a factor:(i) x3+x2+x+1(ii) x4+x3+x2+x+1(iii) x4+3x3+3x2+x+1(iv) x3−x2−(2+2​)x+2​

Text solutionVerified

Practice questions from similar books

Determine which of the following polynomials has $(x + 1)$ a factor:
(i) $x^{3} + x^{2} + x + 1$
(ii) $x^{4} + x^{3} + x^{2} + x + 1$
(iii) $x^{4} + 3 x^{3} + 3 x^{2} + x + 1$
(iv) $x^{3} - x^{2} - (2 + 2) x + 2$