Respuesta :
Answer:
[tex]p (x) = x^{3} - 21x^{2}+ 147x - 343[/tex]
is the required polynomial with degree 3 and p ( 7 ) = 0
Step-by-step explanation:
Given:
p ( 7 ) = 0
To Find:
p ( x ) = ?
Solution:
Given p ( 7 ) = 0 that means substituting 7 in the polynomial function will get the value of the polynomial as 0.
Therefore zero's of the polynomial is seven i.e 7
Degree : Highest raise to power in the polynomial is the degree of the polynomial
We have the identity,
[tex](a -b)^{3} = a^{3}-3a^{2}b +3ab^{2} - b^{3}[/tex]
Take a = x
b = 7
Substitute in the identity we get
[tex](x -7)^{3} = x^{3}-3x^{2}(7) +3x(7)^{2} - 7^{3}\\(x -7)^{3} = x^{3}-21x^{2} +147x - 343[/tex]
Which is the required Polynomial function in degree 3 and if we substitute 7 in the polynomial function will get the value of the polynomial function zero.
p ( 7 ) = 7³ - 21×7² + 147×7 - 7³
p ( 7 ) = 0
[tex]p (x) = x^{3} - 21x^{2}+ 147x - 343[/tex]
Using the Factor Theorem, the polynomial function is given by:
[tex]p(x) = x^3 - 10x^2 + 23x - 14[/tex]
The Factor Theorem states that if a function has roots [tex]x_1, x_2, ..., x_n[/tex], it is written as:
[tex]p(x) = a(x - x_1)(x - x_2)...(x - x_n)[/tex]
In which a is the leading coefficient.
In this problem:
- Degree 3, thus, 3 roots.
- We will consider the leading coefficient to be [tex]a = 1[/tex].
- p(7) = 0, which means that 7 is one root, thus [tex]x_1 = 7[/tex].
- The other two roots, we are going to consider [tex]x_2 = 1, x_3 = 2[/tex].
Then:
[tex]p(x) = (x - 7)(x - 1)(x - 2)[/tex]
[tex]p(x) = (x^2 - 8x + 7)(x - 2)[/tex]
[tex]p(x) = x^3 - 10x^2 + 23x - 14[/tex]
A similar problem is given at https://brainly.com/question/24380382
