Question

The figure shows the curves represented by polynomials f(x), g
g(x), and h(x) of degrees 4, 4, and 2 respectively, on XY plane. Let f(x)-g(x)= ax(x-2)(x-5)(x-9), a ≠0. If b is a negative constant, then choose the most possible expression for h(x) and other correct statements among the given options. (Note that figure is not according to scale .)

h(x)= b(x^2+8x-7)

f(x)=g(x)f(x)=g(x) at x=0,-2,-5,-9

h(x)= b(x^2-6x-7)

h(x)= b(x^2-2x-3)

h(x)= b(x^2-8x+7)

h(x)= b(x^2-6x+7)

f(x)=g(x)f(x)=g(x) at x=0,2,5,9

Answer 1

Answer:

Step-by-step explanation:

h(x)=b(x  2  −6x−7)

f(x)=g(x) at x=0,2,5,9x=0,2,5,9

Answer 2

Concept

This problem is related to the Polynomial which is an algebraic expression that consists of variables and coefficients.

Given

We have a figure below the curves represented by polynomials $f(x), g(x),$ and  $h(x)$  of degrees 4, 4, and 2 respectively, on the X-Y plane. And $f(x)-g(x)= ax(x-2)(x-5)(x-9)$, $a \neq 0$. If b is a negative constant.

To Find

We have to choose the most possible expression for  $h(x)$ and other correct statements among the given options.

1. $h(x)= b(x^2+8x-7)$

2. $f(x)=g(x)$ at  $x=0,-2,-5,-9$

3.  $h(x)= b(x^2-6x-7)$

4. $h(x)= b(x^2-2x-3)$

5. $h(x)= b(x^2-8x+7)$

6. $h(x)= b(x^2-6x+7)$

7. $f(x)=g(x)$ at $x=0,2,5,9$

Solution

We have

$f(x)-g(x)= ax(x-2)(x-5)(x-9)$

As  $a \neq 0$

From here at $x=2, x=5, x=9$

$f(x)=g(x)$

$x=2$  let's put in the equation $h(x)= b(x^2+8x-7)$ we get,

$h(x)= b(2^2+16-7)=b(13)$

As $b < 0$, so, $h(x) < 0$

But from the graph below, $h(x) > 0$

Let us put $x=0$ in the equations $h(x)= b(x^2-2x-3)$ we get,

$h(0)= b(-3)$

$h(x) > 0$

When we put  $x=2$ in the equations $h(x)= b(x^2-2x-3)$ we get,

$h(2)= b(-3)$

$h(x) > 0$

Here, $h(0)=h(2)$ but from the graph of  $h(x)$  it is not possible that both values will be equal.

Let us put $x=0$ in the equations $h(x)= b(x^2-8x+7)$ we get,

$h(0)=b(7)$ here, $h(x) < 0$

But from graph $h(x) > 0$

Now, put $x=0$ in the equations $h(x)= b(x^2-6x+7)$ we get,

$h(0)= b(7)$

$h(x) < 0$

But from graph $h(x) > 0$

Let's put $x=0$ in the equations $h(x)= b(x^2-6x-7)$ we get,

$h(0)= b(-7)$

$h(x) > 0$

When we put  $x=2$ in the equations  $h(x)= b(x^2-6x-7)$  we get,

$h(2)= b(-15)$

$h(x) > 0$

So, $h(x)= b(x^2-6x-7)$ only satisfies the given graph.

As a result, the correct statements among the given options are: $h(x)= b(x^2-6x-7)$  and  $f(x)=g(x)$ at $x=0,2,5,9$

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