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

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comentezq · Accepted Answer

Recall, according to the factor theorem the expression (x–c) 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 (x–0)= is a factor of P(x).  If c = 2 is a zero of P(x) the expression (x–2)= is a factor of P(x).  If c = 4 is a zero of P(x) the expression (x–4)= is a factor of P(x).  2) The polynomial P(x) can be written in factored form as:  P(x) =x(x–4)(x+2)(x–2)  P(x) =(x2–4x)(x+2)(x–2)  3) Apply the difference of squares identity: a2–b2=(a+b)(a–b)  P(x) =(x2–4x)(x2–4)  P(x) =x4–4x3–4x2+16x

Eliza Beth13 · Accepted Answer

Result:f(x)=x4−4x3−4x2+16x or f(x)=(x+2)(x)(x−2)(x−4)Solution:Given the zeros -2, 0, 2, and 4, we can write the factors as follows:(x−(−2))(x−0)(x−2)(x−4)Simplifying each factor, we have:(x+2)(x)(x−2)(x−4)Now, let&#039;s expand this expression:(x+2)(x)(x−2)(x−4)=(x2+2x)(x−2)(x−4)Expanding further:(x2+2x)(x−2)(x−4)=(x2+2x)(x2−2x−4x+8)Combining like terms:(x2+2x)(x2−2x−4x+8)=(x2+2x)(x2−6x+8)Now, we can expand once again:(x2+2x)(x2−6x+8)=x2(x2−6x+8)+2x(x2−6x+8)Expanding further:x2(x2−6x+8)+2x(x2−6x+8)=x4−6x3+8x2+2x3−12x2+16xCombining like terms:x4−6x3+8x2+2x3−12x2+16x=x4−4x3−4x2+16xThus, the polynomial of degree 4 with zeros at -2, 0, 2, and 4 is:f(x)=x4−4x3−4x2+16xYou can also express it in factored form as:f(x)=(x+2)(x)(x−2)(x−4)

Nick Camelot · Accepted Answer

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)(x−0)(x−2)(x−4)

madeleinejames20 · Accepted Answer

Step 1: Let&#039;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(x−r1)(x−r2)(x−r3)(x−r4) 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))(x−0)(x−2)(x−4)Simplifying further:P(x)=a(x+2)(x)(x−2)(x−4)Step 2: Expanding the equation:P(x)=a(x2+2x)(x2−2x−8)Multiplying the terms:P(x)=a(x4−4x3−8x2+16x)Therefore, a polynomial of degree 4 with the given zeros −2,0,2,4 can be expressed as:P(x)=a(x4−4x3−8x2+16x)

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

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