4. Polynomial Functions

Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. √3, -√3, 2, 3

For each polynomial function, find all zeros and their multiplicities. $\text{ƒ}\left(x\right)=(x+1{)}^2(x-1{)}^3(x^2-10)$

For each polynomial function, find all zeros and their multiplicities. $\text{ƒ}\left(x\right)=(x^2+x-2{)}^5(x-1+\sqrt{3}{)}^2$

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. 5+i and 5-i

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=5x²(x²-16)(x+5)

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. f(x)=3x⁴−11x³−x²+19x+6

Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=3x⁵+2x⁴−15x³−10x²+12x+8

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x³+x²−3x+1

Solve: x⁴+2x³−x²−4x−2=0

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x³−3x²−11x+6

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. 2-i, 3, and -1

Determine whether each statement is true or false. If false, explain why. A polynomial function having degree 6 and only real coefficients may have no real zeros.

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)=4x³-37x²+50x+60 between 7 and 8

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴+4x³+6x²+4x+1