4. Polynomial Functions

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. 2x³+x+2; x+1

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)=x³−4x²−7x+10

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

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. 2x⁵+7x⁴−18x²−8x+8=0

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴-6x³+7x²

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

If ƒ(x) is a polynomial function with real coefficients, and if 7+2i is a zero of the function, then what other complex number must also be a zero?

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=4x⁴−x³+5x²−2x−6

Determine whether each statement is true or false. If false, explain why. Because x-1 is a factor of ƒ(x)=x⁶-x⁴+2x²-2, we can also conclude that ƒ(1) = 0.

Solve each problem. Use Descartes' rule of signs to determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of $\text{ƒ}\left(x\right)=x^3+3x^2-4x-2$.

In Exercises 47–48, find an nth-degree polynomial function with real coefficients satisfying the given conditions. Verify the real zeros and the given function value. n = 3; 2 and 2 - 3i are zeros; f(1) = -10

Determine whether each statement is true or false. If false, explain why. The product of a complex number and its conjugate is always a real number.

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=5x³−3x²+3x−1

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴+2x²+1

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