For the given statements select the correct option.

Assertion : $x = 1$ is zero of the polynomial for $p (x) = x^{3} + x^{2} - 1$

Reason : $p (1) = 1$

Assertion is true but reason is false.

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Assertion is false but reason is true

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Both assertion and reason are true but reason is not the correct explanation of assertion.

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Both assertion and reason are true and reason is the correct explanation of assertion.

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Solution

The correct option is B Assertion is false but reason is true
We know that,
If $x = a$ is zero of the polynomial for $p (x)$ then $p (a) = 0$

Now, considering $p (x) = x^{3} + x^{2} - 1$ and equating $x = 1$ in $p (x)$

We get, $p (1) = 1^{3} + 1^{2} - 1$
$p (1) = 1$

Hence, Assetion is false and reason is true.

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Similar questions

Q. Read the statements given below:
I. If α, β are the zeros of the polynomial x² − p(x + 1) −c, then (a + 1)(β + 1) = 1 − c.

II. If α, β are the zeros of the polynomial x² + px + q, then the polynomial having

\frac{1}{α}, \frac{1}{β}

as zeros is qx² + px + 1.
III. When x³+ 3x² − 5x + 4 is divided by (x + 1), then the remainder is 9.

Which of the above statements is false?

(a) I only
(b) II only
(c) III only
(d) I and III both

Now for each pair of polynomials p(x) and q(x) given below, find p(x) + q(x) and p(x) − q(x). Also compute p(1) + q(1) and p(1) − q(1) for each.

(i)

(ii)

(iii)

(iv)

(v)

Q. Verify that:
(i) 4 is a zero of the polynomial p(x) = x − 4.
(ii) −3 is a zero of the polynomial p(x) = x − 3.
(iii)

- \frac{1}{2}

is a zero of the polynomial p(y) = 2y + 1.
(iv)

\frac{2}{5}

is a zero of the polynomial p(x) = 2 − 5x.
(v) 1 and 2 are the zeros of the polynomial p(x) = (x − 1)(x − 2).
(vi) 0 and 3 are the zeros of the polynomial p(x) = x² − 3x.
(vii) 2 and −3 are the zeros of the polynomial p(x) = x² + x − 6.

From the polynomial p(x) = x² + x + 1, do we get p(x) = 0 for any x? What about p(x) = 1? And p(x) = −1?

For the given statements select the correct option.

Assertion (A): If on dividing the polynomial $p (x) = x^{2} - 3 a x + 3 a - 7$ by $(x + 1)$ , we get 6 as remainder, then $a = 3$ .

Reason (R): When a polynomial $p (x)$ is divided by $(x - a)$ , then the remainder is $p (a)$ .

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For the given statements select the correct option. Assertion : x=1 is zero of the polynomial for p(x)=x3+x2−1 Reason : p(1)=1

The correct option is B Assertion is false but reason is trueWe know that, If x=a is zero of the polynomial for p(x) then p(a)=0 Now, considering p(x)=x3+x2−1 and equating x=1 in p(x) We get, p(1)=13+12−1 p(1)=1 Hence, Assetion is false and reason is true.

For the given statements select the correct option.

Assertion : $x = 1$ is zero of the polynomial for $p (x) = x^{3} + x^{2} - 1$

Reason : $p (1) = 1$