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Polynomials and Rational Theorem Algebra Questions

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1. Find a bound on the real zeros of the polynomial function f(x)=x4+x3-4x-6
Every real zero of f lies between -2______ and____2__.
2. Use the intermediate value theorem to show that the polynomial function has a zero in the given interval.
f(x)=11x4-4x2+6x-1;[0,3] enter the value of f (0)
f(0)=___-1___ (simplify)
Reason: f(0)=-1 and f(3)=872 which have different sign, by IVT, there is a number c between 0 and 3 such that f(c)=0
3. Use the intermediate value theorem to show that the polynomial function has a zero in the given interval.
f(x)=5x3+7x2-9x+7;[-4,-1] find the value of f (-4).
f(-4)=_-250__
Reason: f(-4)=-250 and f(-1)=18 which have different sign, by IVT, there is a number c between -4 and -1 such that f(c)=0

4. Form a polynomial f(x) with real coefficents having the given degree and zeros
Degree 5; Zeros: 2; -i;-7+i
Enter the polynomial f(x)=a(____) type expression using x as the variable
Reason: since -i is zero, i must be zero too. Since -7+i is zero, -7-i must be zero too. So
f(x)=(x-2)(x+i)(x-i))(x+7-i)(x+7+i)=(x-2)(x^2+1)(x^2+14x+50)

5. Find the complex zero of the polynomial function. Write f in factored form.
f(x)=x3-5x2+16x-30
Use the complex zeros to write f in factored form f(x)=_____ (reduce fractions)
Reason: first we find that x=3 happen to be the real root. After division, f(x)=(x-3)(x^2-2x+10).
So f(x)=(x-3)(x-1+3i)(x-1-3i). So complex zeros are 1-3i and 1+3i

6. Use the rational theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.
f(x)=x3-4x2-19x-14 x=__-1,-2,7____
Reason: by rational theorem. p=-14 and q=1. So all possible real roots are 1,-1,2,-2,7,-7,14,-14.
So after trying all roots, there are real 3 roots of f(x): -1, -2, 7

7. Use the rational theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.
f(x)=2x3-x2+2x-1 x=__1/2____

Reason: by rational theorem. p=-1 and q=2. So all possible roots are 1/2 and -1/2.
So after trying all roots, there are 1 real roots of f(x): 1/2.

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