Respuesta :
We are given zeros : (−1,0), (−5,0) and (−7,0) .
We can write those zeros as
x=-1, x=-5 and x=-7.
So, the factors of the polynomial would be
(x+1) , (x+5) and (x+7).
We also given leading coefficent =1.
Let us multiply all those factors of the polynomila we got and then multiply by 1 finally.
(x+1) * (x+5) * (x+7).
Let us foil first two factors (x+1) * (x+5) first, we get
x^2 + 5x + 1x + 5 = x^2 + 6x +5.
Therefore, (x+1) * (x+5) * (x+7) = (x^2 + 6x +5 ) (x+7).
Let us multiply (x^2 + 6x +5 ) and (x+7), we get
(x^2 + 6x +5 ) * (x+7) = x^3 + 7x^2 + 6x^2 + 42x + 5x + 35.
Combining like terms 7x^2 + 6x^2 = 13x^2 and 42x + 5x = 47x.
Therefore,
x^3 + 7x^2 + 6x^2 + 42x + 5x + 35
= x^3 +13x^2 +47x +35.
If we multiply it by 1, we get same terms.
And also if we plug x=0, we get
(0)^3 +13(0)^2 +47(0) +35 = 0+0+0+35 = 35.
We can see that y-intercept is 35 there, because we get 35 on plugging x=0 .
Therefore, x^3 +13x^2 +47x +35 is the polynomial in standard form that fulfil all the given requirements.
Using the Factor Theorem, the polynomial is given by:
[tex]p(x) = x^3 + 13x^2 + 47x + 35[/tex]
The Factor Theorem states that if a polynomial has roots [tex]x_1, x_2, ..., x_n[/tex], it can be written as:
[tex]p(x) = a(x - x_1)(x - x_2)(x - 3)[/tex]
In which a is the leading coefficient.
In this problem:
- Leading coefficient of 1, thus [tex]a = 1[/tex].
- Zeros of [tex]x_1 = -1, x_2 = -5, x_3 = -7[/tex].
Then:
[tex]p(x) = (x + 1)(x + 5)(x + 7)[/tex]
[tex]p(x) = (x^2 + 6x + 5)(x + 7)[/tex]
[tex]p(x) = x^3 + 13x^2 + 47x + 35[/tex]
A similar problem is given at https://brainly.com/question/24380382
