Studdy Solution
STEP 1
1. The polynomial f(x) has real coefficients.
2. The polynomial is of degree 5.
3. The given zeros are 8, −i, and −3+i.
4. Complex zeros occur in conjugate pairs when coefficients are real.
STEP 2
1. Identify all zeros of the polynomial.
2. Write the polynomial in factored form using the zeros.
3. Expand the factored form to obtain the polynomial in standard form.
STEP 3
Identify all zeros of the polynomial:
- Given zeros are 8, −i, and −3+i.
- Since the polynomial has real coefficients, the complex zeros must include their conjugates.
- The conjugate of −i is i.
- The conjugate of −3+i is −3−i.
Thus, the zeros are 8,−i,i,−3+i,−3−i.
STEP 4
Write the polynomial in factored form using the zeros:
f(x)=(x−8)(x+i)(x−i)(x+3−i)(x+3+i)
STEP 5
Expand the factored form to obtain the polynomial in standard form:
First, simplify the pairs of complex conjugates:
(x+i)(x−i)=x2+1
(x+3−i)(x+3+i)=(x+3)2−i2=(x+3)2+1
Now expand (x+3)2+1:
(x+3)2=x2+6x+9
(x+3)2+1=x2+6x+9+1=x2+6x+10
Now, the polynomial becomes:
f(x)=(x−8)(x2+1)(x2+6x+10)
Expand the expression:
First, expand (x2+1)(x2+6x+10):
(x2+1)(x2+6x+10)=x4+6x3+10x2+x2+6x+10
Combine like terms:
=x4+6x3+11x2+6x+10
Now, multiply by (x−8):
f(x)=(x−8)(x4+6x3+11x2+6x+10)
Expand this product:
=x(x4+6x3+11x2+6x+10)−8(x4+6x3+11x2+6x+10)
=x5+6x4+11x3+6x2+10x−8x4−48x3−88x2−48x−80
Combine like terms:
=x5−2x4−37x3−82x2−38x−80
The polynomial f(x) is:
f(x)=x5−2x4−37x3−82x2−38x−80
