4. Polynomial Functions

Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. -2+√5, -2-√5, -2, 1

In Exercises 25–32, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=4; i and 3i are zeros; f(-1) = 20

Find all rational zeros of each function. ƒ(x)=8x^4-14x^3-29x^2-4x+3

Factor ƒ(x) into linear factors given that k is a zero. See Example 2. ƒ(x)=x^4+2x^3-7x^2-20x-12; k=-2 (multiplicity 2)

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2between -1 and 0

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2between 2 and 3

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2Find the zero in part (b) to three decimal places.

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x^3-37x^2+50x+60between 7 and 8

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x^3-37x^2+50x+60between 2 and 3

In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=2x^4−5x^3−x^2−6x+4

For each polynomial function, one zero is given. Find all other zeros. See Examples 2 and 6. ƒ(x)=-x^4-5x^2-4; -i

For Exercises 40–46,(a) List all possible rational roots or rational zeros.(b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros.(c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero.(d) Use the quotient from part (c) to find all the remaining roots or zeros.f(x) = x^3 + 3x^2 - 4

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. 4x^4−x^3+5x^2−2x−6=0

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(2x^2-7x+3)^3(x-2-√5)

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. See Example 4. Zeros of -3, 1, and 4; ƒ(2)=30