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Factorisation of Quadratic Polynomials

If x - 1, x...

Question

If $x - 1, x - 3$ and $x + 1$ are all factors of a polynomial $P (x)$ of degree $3$ , which of the following must also be a factor of $P (x)$ ?
I) $x^{2} + 1$
II) $x^{2} - 1$
III) $x^{2} - 4 x + 3$

A

II and III only

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B

I and II only

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C

III only

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D

II only

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E

I only

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Solution

The correct option is A II and III only
$P (x) = k (x - 1) (x - 3) (x + 1)$
factors can be $(x^{2} - 1), (x^{2} - 4 x + 3)$

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Similar questions

Q. Question 3
Verify whether the following are zeroes of the polynomial, indicated against them.
(i) p(x) = 3x + 1, x =

- \frac{1}{3}

p (x) = 5 x - π, x = \frac{4}{5}

p (x) = x^{2} - 1, x = 1, - 1

(iv) p(x) = (x +1) (x -2), x = - 1, 2
(v)

p (x) = x^{2}, x = 0

p (x) = i x + m, x = - \frac{m}{i}

p (x) = 3 x^{2} - 1, x = - \frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}

p (x) = 2 x + 1, x = \frac{1}{2}

Q. Classify the following polynomials based on their degree.
(i)

p (x) = 3

p (y) = \frac{5}{2} y^{2} + 1

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

p (x) = 3 x^{2}

p (x) = x + 3

p (x) = - 7

p (x) = x^{3} + 1

p (x) = 5 x^{2} - 3 x + 2

p (x) = 4 x

p (x) = \frac{3}{2}

p (x) = \sqrt{3} x + 1

p (y) = y^{3} + 3 y

Q. Use the Factor Theorem to determine whether

g (x)

p (x)

in each of the following cases:
(i)

p (x) = 2 x^{3} + x^{2} - 2 x - 1, g (x) = x + 1

p (x) = x^{3} + 3 x^{2} + 3 x + 1, g (x) = x + 2

p (x) = x^{3} - 4 x^{2} + x + 6, g (x) = x - 3

Q. Divide the polynomial

p (x)

by the polynomial

g (x)

and find the quotient and remainder in each of the following:
(i)

p (x) = x^{3} - 3 x^{2} + 5 x - 3, g (x) = x^{2} - 2

p (x) = x^{4} - 3 x^{2} + 4 x + 5, g (x) = x^{2} + 1 - x

p (x) = x^{4} - 5 x + 6, g (x) = 2 - x^{2}

Q. A polynomial

p (x)

g (x)

, the obtained quotient

q (x)

and the remainder

r (x)

are given in the table. Find

p (x)

Sl.	$p (x)$	$g (x)$	$q (x)$	$r (x)$
i	?	$x - 2$	$x^{2} - x + 1$	$4$
ii	?	$x + 3$	$2 x^{2} + x + 5$	$3 x + 1$
iii	?	$2 x + 1$	$x^{3} + 3 x^{2} - x + 1$	$0$
iv	?	$x - 1$	$x^{3} - x^{2} - x - 1$	$2 x - 4$
v	?	$x^{2} + 2 x + 1$	$x^{4} - 2 x^{2} + 5 x - 7$	$4 x + 12$

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