4. Polynomial Functions

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

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

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $\text{ƒ}\left(x\right)=-8x^4+3x^3-6x^2+5x-7$

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. 4x⁴−x³+5x²−2x−6=0

Solve each problem. Give the maximum number of turning points of the graph of each function. ƒ(x)=4x^3-6x^2+2

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $\text{ƒ}\left(x\right)=2x^5-7x^3+6x+8$

For each polynomial function, one zero is given. Find all other zeros. $\text{ƒ}\left(x\right)=x^3+4x^2-5;1$

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x²+4x+6=0

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

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

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $\text{ƒ}\left(x\right)=-2x^5+10x^4-6x^3+8x^2-x+1$

Find all rational zeros of each function. $\text{ƒ}\left(x\right)=8x^4-14x^3-29x^2-4x+3$

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴-8x³+29x²-66x+72

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=4x³+3x²+8x+6

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)=x³−2x²−11x+12