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(A) To determine the number of points where the graph of a polynomial $p(x)$ intersects the $X$-axis,we find the number of real roots of the equation

Vedclass · Accepted Answer

$(1-c), (2-a), (3-d), (4-b)$

Vedclass · Answer

(A) To determine the number of points where the graph of a polynomial $p(x)$ intersects the $X$-axis,we find the number of real roots of the equation $p(x) = 0$.$1.$ $p(x) = x^{2}-5x+6 = (x-2)(x-3)$. The roots are $x=2, 3$. It intersects the $X$-axis at two points. So,$1-c$.$2.$ $p(x) = x^{2}-10x+25 = (x-5)^{2}$. The root is $x=5$ (repeated). It intersects the $X$-axis at one point. So,$2-a$.$3.$ $p(x) = x^{3}-4x = x(x^{2}-4) = x(x-2)(x+2)$. The roots are $x=0, 2, -2$. It intersects the $X$-axis at three points. So,$3-d$.$4.$ $p(x) = x^{2}+4x+5$. The discriminant $D = b^{2}-4ac = 16 - 4(1)(5) = 16 - 20 = -4$. Since $D Thus,the correct matching is $(1-c), (2-a), (3-d), (4-b)$.

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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