Question

Help me. The graph of a polynomial with unknown a is given. Write a polynomial

Answer 1

Answer:

$f(x)=\dfrac{7}{20}\left(x+2\right)^{2}\left(x-1\right)$.

Step-by-step explanation:

If a graph touches the x-axis at x=c, then (x-c) is a factor of function.

From the given graph it is clear that the graph of function intersect x-axis at x=1 and touch the x-axis at x=-2.

It means (x-1) and (x+2) are not factors of given function but power of (x+2) must be 2.

So, the required function is

$f(x)=a(x+2)^2(x-1)$    ...(1)

where, a is a constant.

From the given figure it is clear that the graph passes through the point (-4,-7). So, substitute x=-4 and f(x)=-7 in the above function.

$-7=a(-4+2)^2(-4-1)$

$-7=a(-2)^2(-5)$

$-7=-20a$

$\dfrac{7}{20}=a$

Substitute $a=\dfrac{7}{20}$ in (1).

$f(x)=\dfrac{7}{20}(x+2)^2(x-1)$

Therefore, the required function is $f(x)=\dfrac{7}{20}(x+2)^2(x-1)$.