Find a polynomial of the specified degree that has the given zeros. Degree 4, zeros -2, 0, 2, 4

Nann

Nann

Answered question

2021-02-25

Find a polynomial of the specified degree that has the given zeros. Degree 4, zeros -2, 0, 2, 4

Answer & Explanation

comentezq

comentezq

Skilled2021-02-26Added 106 answers

Recall, according to the factor theorem the expression (xc) is a factor of a polynomial P(x) if and only if P(c) = 0.
1) Let a polynomial function P(x):
If c = -2 is a zero of P(x) the expression (x+2) is a factor of P(x).
If c = 0 is a zero of P(x) the expression (x0)= is a factor of P(x).
If c = 2 is a zero of P(x) the expression (x2)= is a factor of P(x).
If c = 4 is a zero of P(x) the expression (x4)= is a factor of P(x).
2) The polynomial P(x) can be written in factored form as:
P(x) =x(x4)(x+2)(x2)
P(x) =(x24x)(x+2)(x2)
3) Apply the difference of squares identity: a2b2=(a+b)(ab)
P(x) =(x24x)(x24)
P(x) =x44x34x2+16x
Eliza Beth13

Eliza Beth13

Skilled2023-06-19Added 130 answers

Result:
f(x)=x44x34x2+16x or f(x)=(x+2)(x)(x2)(x4)
Solution:
Given the zeros -2, 0, 2, and 4, we can write the factors as follows:
(x(2))(x0)(x2)(x4)
Simplifying each factor, we have:
(x+2)(x)(x2)(x4)
Now, let's expand this expression:
(x+2)(x)(x2)(x4)=(x2+2x)(x2)(x4)
Expanding further:
(x2+2x)(x2)(x4)=(x2+2x)(x22x4x+8)
Combining like terms:
(x2+2x)(x22x4x+8)=(x2+2x)(x26x+8)
Now, we can expand once again:
(x2+2x)(x26x+8)=x2(x26x+8)+2x(x26x+8)
Expanding further:
x2(x26x+8)+2x(x26x+8)=x46x3+8x2+2x312x2+16x
Combining like terms:
x46x3+8x2+2x312x2+16x=x44x34x2+16x
Thus, the polynomial of degree 4 with zeros at -2, 0, 2, and 4 is:
f(x)=x44x34x2+16x
You can also express it in factored form as:
f(x)=(x+2)(x)(x2)(x4)
Nick Camelot

Nick Camelot

Skilled2023-06-19Added 164 answers

To find a polynomial of degree 4 with zeros -2, 0, 2, and 4, we can use the factored form of a polynomial. The factored form is obtained by multiplying the factors (x - zero) for each zero of the polynomial.
Therefore, the polynomial is given by:
P(x)=(x+2)(x0)(x2)(x4)
madeleinejames20

madeleinejames20

Skilled2023-06-19Added 165 answers

Step 1: Let's denote the polynomial as P(x), and we know that it has four zeros. Therefore, we can write the polynomial as:
P(x)=a(xr1)(xr2)(xr3)(xr4) where a is a constant and r1,r2,r3,r4 are the zeros 2,0,2,4, respectively.
Substituting these zeros into the equation, we get:
P(x)=a(x(2))(x0)(x2)(x4)
Simplifying further:
P(x)=a(x+2)(x)(x2)(x4)
Step 2: Expanding the equation:
P(x)=a(x2+2x)(x22x8)
Multiplying the terms:
P(x)=a(x44x38x2+16x)
Therefore, a polynomial of degree 4 with the given zeros 2,0,2,4 can be expressed as:
P(x)=a(x44x38x2+16x)

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