Question

Find a formula for the given polynomial.

Answer 1

In this question, we have to identify the zeros of the polynomial, along with a point, and then we get that the formula for the polynomial is:

$p(x) = -0.5(x^3 - x^2 + 6x)$

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Equation of a polynomial, according to it's zeros:

Given a polynomial f(x), this polynomial has roots such that it can be written as: , in which a is the leading coefficient.

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Identifying the zeros:

Given the graph, the zeros are the points where the graph crosses the x-axis. In this question, they are:

$x_1 = -2, x_2 = 0, x_3 = 3$

Thus

$p(x) = a(x - x_{1})(x - x_{2})(x-x_3)$

$p(x) = a(x - (-2))(x - 0)(x-3)$

$p(x) = ax(x+2)(x-3)$

$p(x) = ax(x^2 - x + 6)$

$p(x) = a(x^3 - x^2 + 6x)$

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Leading coefficient:

Passes through point (2,-8), that is, when $x = 2, y = -8$, which is used to find a. So

$p(x) = a(x^3 - x^2 + 6x)$

$-8 = a(2^3 - 2^2 + 6*2)$

$16a = -8$

$a = -\frac{8}{16} = -0.5$

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Considering the zeros and the leading coefficient, the formula is:

$p(x) = -0.5(x^3 - x^2 + 6x)$

A similar problem is found at brainly.com/question/16078990

Answer 2

The formula that represents the polynomial in the figure is $p(x) = x^{3}-x^{2}-6\cdot x$.

Based on the Fundamental Theorem of Algebra, we understand that Polynomials with real Coefficient have at least one real Root and at most a number of Roots equal to its Grade. The Grade is the maximum exponent that Polynomial has and root is a point such that $p(x) = 0$. By Algebra we understand that polynomial can be represented in this manner known as Factorized form:

$p(x) = \Pi\limits_{i=0}^{n} (x-r_i)$ (1)

Where:

$n$ - Grade of the polynomial.

$i$ - Index of the root binomial.

$x$ - Independent variable.

We notice that polynomials has three roots in $x = -2$, $x = 0$ and $x = 3$, having the following construction:

$p(x) =(x+2)\cdot x \cdot (x-3)$

$p(x) = (x^{2}+2\cdot x)\cdot (x-3)$

$p(x) = x^{3}+2\cdot x^{2}-3\cdot x^{2}-6\cdot x$

$p(x) = x^{3}-x^{2}-6\cdot x$

The formula that represents the polynomial in the figure is $p(x) = x^{3}-x^{2}-6\cdot x$.

Here is a question related to the determination polynomials: brainly.com/question/10241002