4. Polynomial Functions

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?

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=3; -5 and 4+3i are zeros; f(2) = 91

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)=6x^4+13x^3-11x^2-3x+5 no zero less than -3

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

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $f\left(x\right)=x^4-6x^3+14x^2-14x+5$

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

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. $f\left(x\right)=2x^4+3x^3+3x-2$

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

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x²+4x−1=0

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $\text{ƒ}\left(x\right)=3x^3+6x^2+x+7$

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $\text{ƒ}\left(x\right)=x^5+3x^4-x^3+2x+3$

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=9x⁶-7x⁴+8x²+x+6

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $\text{ƒ}\left(x\right)=7x^5+6x^4+2x^3+9x^2+x+5$

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $\text{ƒ}\left(x\right)=2x^5+11x^4+16x^3+15x^2+36x$

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.