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Standard X

Mathematics

Division Algorithm for a Polynomial

What is the H...

Question

What is the HCF of the polynomials $x^{4} - 3 x + 2, x^{3} - 3 x^{2} + 3 x - 1$ and $x^{4} - 1$ ?

x - 1

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x + 2

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x^{2} - 1

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None of the above

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Solution

The correct option is B $x - 1$
$x^{4} - 3 x + 2 = x^{4} - x^{3} + x^{3} - x^{2} + x^{2} - x - 2 x + 2$
$= x^{3} (x - 1) + x^{2} (x - 1) + x (x - 1) - 2 (x - 1)$
$= (x - 1) [x^{3} + x^{2} + x - 2]$
$x^{3} - 3 x^{2} + 3 x - 1 = (x - 1)^{3}$
$x^{4} - 1 = (x - 1) (x + 1) (x^{2} + 1)$
$H C F = x - 1$

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Similar questions

Q. Which of the following polynomials has (x + 1) as 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

II.

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

Q. 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 + \sqrt{2}) x + \sqrt{2}

Q. Question 1
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 + \sqrt{2}) x + \sqrt{2}

Now in each of the pairs of polynomials given below, check whether the first is a factor of the second:

(i) x + 1, x³ − 1

(ii) x − 1, x³ + 1

(iii) x + 1, x³ + 1

(iv) x² − 1, x⁴ − 1

(v) x − 1, x⁴ − 1

(vi) x + 1, x⁴ − 1

(vii) x − 2, x² − 5x + 1

(viii) x + 2, x² + 5x + 6

(ix)

(x) 1.3x − 2.6, x² − 5x + 6

Q. Question 1
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 + \sqrt{2}) x + \sqrt{2}

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What is the HCF of the polynomials x4−3x+2,x3−3x2+3x−1 and x4−1?

The correct option is B x−1x4−3x+2=x4−x3+x3−x2+x2−x−2x+2=x3(x−1)+x2(x−1)+x(x−1)−2(x−1)=(x−1)[x3+x2+x−2]x3−3x2+3x−1=(x−1)3x4−1=(x−1)(x+1)(x2+1)HCF=x−1

What is the HCF of the polynomials $x^{4} - 3 x + 2, x^{3} - 3 x^{2} + 3 x - 1$ and $x^{4} - 1$ ?

The correct option is B $x - 1$
$x^{4} - 3 x + 2 = x^{4} - x^{3} + x^{3} - x^{2} + x^{2} - x - 2 x + 2$
$= x^{3} (x - 1) + x^{2} (x - 1) + x (x - 1) - 2 (x - 1)$
$= (x - 1) [x^{3} + x^{2} + x - 2]$
$x^{3} - 3 x^{2} + 3 x - 1 = (x - 1)^{3}$
$x^{4} - 1 = (x - 1) (x + 1) (x^{2} + 1)$
$H C F = x - 1$