4. Polynomial Functions

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

Find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. $f\left(x\right)=-2(x-1)(x+2{)}^2(x+5{)}^3$

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x³−10x−12=0

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. $2x^4+5x^3-2x^2+5x+6;x+3$

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

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

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. $\text{ƒ}\left(x\right)=8x^4-14x^3-29x^2-4x+3$

Factor $\text{ƒ}\left(x\right)$ into linear factors given that k is a zero. $\text{ƒ}\left(x\right)=x^4+2x^3-7x^2-20x-12;k=-2$ (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+2 between -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+2 between 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+2 Find 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)=4x³-37x²+50x+60 between 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)=4x³-37x²+50x+60 between 2 and 3

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=2x⁴−5x³−x²−6x+4