Mix Examples - Polynomials Questions in English

(C) The number of real zeros of a polynomial $p(x)$ is equal to the number of points where its graph $y=p(x)$ intersects the $X$-axis.
In the given figure,the curve $y=p(x)$ intersects the $X$-axis at two distinct points.
Therefore,the polynomial $p(x)$ has $2$ real zeros.

(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 $D < 0$,there are no real roots. It intersects the $X$-axis at zero points. So,$4-b$.
Thus,the correct matching is $(1-c), (2-a), (3-d), (4-b)$.