Math  /  Algebra

QuestionForm a polynomial f(x)f(x) with real coefficients having the given degree and zeros. Degree 4;4 ; zeros: 5 , multiplicity 2;4i2 ; 4 i

Studdy Solution

STEP 1

1. The polynomial f(x) f(x) is of degree 4.
2. The zeros of the polynomial are given as 5 with multiplicity 2, and 4i 4i .
3. Since the polynomial has real coefficients, the complex zero 4i 4i must have its conjugate 4i-4i as another zero.

STEP 2

1. Identify all zeros of the polynomial, including multiplicities and complex conjugates.
2. Write the polynomial in factored form using the identified zeros.
3. Expand the factored form to obtain the polynomial in standard form.

STEP 3

Identify all zeros of the polynomial:
- Zero 1: x=5 x = 5 with multiplicity 2, so it appears twice. - Zero 2: x=4i x = 4i - Zero 3: x=4i x = -4i (the conjugate of 4i 4i )

STEP 4

Write the polynomial in factored form using the zeros:
The polynomial can be expressed as:
f(x)=(x5)2(x4i)(x+4i) f(x) = (x - 5)^2 (x - 4i)(x + 4i)

STEP 5

Expand the factors to obtain the polynomial in standard form:
First, expand the complex conjugate pair:
(x4i)(x+4i)=x2(4i)2=x2(16)=x2+16 (x - 4i)(x + 4i) = x^2 - (4i)^2 = x^2 - (-16) = x^2 + 16
Now, substitute back into the polynomial:
f(x)=(x5)2(x2+16) f(x) = (x - 5)^2 (x^2 + 16)
Expand (x5)2 (x - 5)^2 :
(x5)2=x210x+25 (x - 5)^2 = x^2 - 10x + 25
Substitute back into the polynomial:
f(x)=(x210x+25)(x2+16) f(x) = (x^2 - 10x + 25)(x^2 + 16)
Now, expand the product:
f(x)=x2(x2+16)10x(x2+16)+25(x2+16) f(x) = x^2(x^2 + 16) - 10x(x^2 + 16) + 25(x^2 + 16)
f(x)=x4+16x210x3160x+25x2+400 f(x) = x^4 + 16x^2 - 10x^3 - 160x + 25x^2 + 400
Combine like terms:
f(x)=x410x3+41x2160x+400 f(x) = x^4 - 10x^3 + 41x^2 - 160x + 400
The polynomial f(x) f(x) in standard form is:
f(x)=x410x3+41x2160x+400 f(x) = x^4 - 10x^3 + 41x^2 - 160x + 400

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