(A) To find the zeros of the polynomial $p(x) = x^3 - x$,we set $p(x) = 0$.$x^3 - x = 0$Factor out $x$:$x(x^2 - 1) = 0$Using the difference of squares

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(A) To find the zeros of the polynomial $p(x) = x^3 - x$,we set $p(x) = 0$.$x^3 - x = 0$Factor out $x$:$x(x^2 - 1) = 0$Using the difference of squares formula $a^2 - b^2 = (a - b)(a + b)$:$x(x - 1)(x + 1) = 0$Setting each factor to zero,we get:$x = 0$,$x - 1 = 0 \implies x = 1$,and $x + 1 = 0 \implies x = -1$.Thus,the zeros are $0, 1, -1$.There are $3$ real zeros.

Class 10 Mathematics Polynomials Geometrical Meaning of the Zeroes of a Polynomial

Medium

The number of real zeros of the polynomial $p(x) = x^3 - x$ is . . . . . . .

A
$3$
B
$1$
C
$2$
D
$0$

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Geometrical Meaning of the Zeroes of a Polynomial

The graph of a linear polynomial $p(x) = ax + b$ intersects the $X$-axis at $\ldots \ldots \ldots \ldots$ (where $a \neq 0$).

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Which of the following groups matches the data of Part $I$ with the data of Part $II$?

Part $I$ Part $II$

$1.$ The graph of $p(x)=x^{2}-5x+6$ $a.$ Intersects the $X$-axis at one point.

$2.$ The graph of $p(x)=x^{2}-10x+25$ $b.$ Intersects the $X$-axis at zero points.

$3.$ The graph of $p(x)=x^{3}-4x$ $c.$ Intersects the $X$-axis at two points.

$4.$ The graph of $p(x)=x^{2}+4x+5$ $d.$ Intersects the $X$-axis at three points.

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The number of real zeros of $y=p(x)$ is ............... in the given figure.

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Find the number of real zeros of the following polynomial: $p(x) = 5x - 4$.

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Are the following statements 'True' or 'False'? Justify your answers.
If the graph of a polynomial intersects the $x$-axis at only one point,it cannot be a quadratic polynomial.

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Part $I$	Part $II$
$1.$ The graph of $p(x)=x^{2}-5x+6$	$a.$ Intersects the $X$-axis at one point.
$2.$ The graph of $p(x)=x^{2}-10x+25$	$b.$ Intersects the $X$-axis at zero points.
$3.$ The graph of $p(x)=x^{3}-4x$	$c.$ Intersects the $X$-axis at two points.
$4.$ The graph of $p(x)=x^{2}+4x+5$	$d.$ Intersects the $X$-axis at three points.

The number of real zeros of the polynomial $p(x) = x^3 - x$ is . . . . . . .

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The graph of a linear polynomial $p(x) = ax + b$ intersects the $X$-axis at $\ldots \ldots \ldots \ldots$ (where $a \neq 0$).

The number of real zeros of $y=p(x)$ is ............... in the given figure.

Find the number of real zeros of the following polynomial: $p(x) = 5x - 4$.

Are the following statements 'True' or 'False'? Justify your answers.If the graph of a polynomial intersects the $x$-axis at only one point,it cannot be a quadratic polynomial.

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Are the following statements 'True' or 'False'? Justify your answers.
If the graph of a polynomial intersects the $x$-axis at only one point,it cannot be a quadratic polynomial.