1. Review of Algebra

2. Equations & Inequalities

3. Functions & Graphs

4. Polynomial Functions

5. Rational Functions

6. Exponential and Logarithmic Functions

7. Systems of Equations and Inequalities

8. Matrices and Determinants

9. Conic Sections

10. Sequences, Induction, and Probability

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In Exercises 25–32, 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

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. See Example 4.
Zero of -3 having multiplicity 3; ƒ(3)=36

Finding all the Zeros of a Polynomial - Example 2

Solve each problem. Is x+1 a factor of ƒ(x)=x^3+2x^2+3x+2?

For Exercises 40–46,

(a) List all possible rational roots or rational zeros.
(b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros.
(c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero.
(d) Use the quotient from part (c) to find all the remaining roots or zeros.


f(x) = x^3 + 3x^2 - 4

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=3x(x-2)(x+3)(x^2-1)

Find the Equation from the Graph of a Polynomial

For each polynomial function, one zero is given. Find all other zeros. See Examples 2 and 6. ƒ(x)=4x^3+6x^2-2x-1; 1/2

Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x^4−3x^2−4). If f(3)=−150, determine the value of a_n.

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=2x^5+11x^4+16x^3+15x^2+36x

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=5x^3-9x^2+28x+6

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. x^3+6x^2-2x-7; x+1

Factor ƒ(x) into linear factors given that k is a zero. See Example 2. ƒ(x)=2x^3-3x^2-5x+6; k=1

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=x^5+3x^4-x^3+2x+3

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^4+2x^3-3x^2+24x-180

Descartes' Rule of Signs

For each polynomial function, one zero is given. Find all other zeros. See Examples 2 and 6. ƒ(x)=x^3-x^2-4x-6; 3

Show that f(x) = x^3 - 2x - 1 has a real zero between 1 and 2.

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(x+1)^2(x-1)^3(x^2-10)

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(x^2+x-2)^5(x-1+√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. See Examples 4–6. 
5+i and 5-i

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=5x^2(x^2-16)(x+5)

Use descartes rule of signs to find the number of positive and negative real zeros

Write the equation of a polynomial using its x-intercepts

In Exercises 39–52, 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)=3x^4−11x^3−x^2+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)=3x^5+2x^4−15x^3−10x^2+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)=2x^3+x^2−3x+1

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)=2x^3−3x^2−11x+6

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. See Examples 4–6.
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)=4x^3-37x^2+50x+60

between 7 and 8

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^4+4x^3+6x^2+4x+1

In Exercises 35–36, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x) = x^4 - 6x^3 + 14x^2 -14x + 5

In Exercises 25–32, 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; 1 and 5i are zeros; f(-1) = -104

Solve each problem. Find a polynomial function ƒ of degree 3 with -2, 1, and 4 as zeros, and ƒ(2)=16.

In Exercises 39–52, 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. 2x^3−x^2−9x−4=0

The Rational Zeros Theorem

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=7x^5+6x^4+2x^3+9x^2+x+5

For each polynomial function, one zero is given. Find all other zeros. See Examples 2 and 6. ƒ(x)=-x^4-5x^2-4; -i

In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=2x^4−5x^3−x^2−6x+4

Solving Polynomial Equations By Factoring and Using Synthetic Division

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=5x^6-6x^5+7x^3-4x^2+x+2

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. x^3-5x^2+3x+1; x-1

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^4+x^3-9x^2+11x-4

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x^4−11x^3−x^2+19x+6

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 greater than 1

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)=x^3+x^2−4x−4

Use Descartes' Rule of Signs to explain why 2x^4 + 6x^2 + 8 = 0 has no real roots.

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. See Example 4. 
Zeros of -2, 1, and 0; ƒ(-1)=-1

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)=−x^3+x^2+16x−16

In Exercises 25–32, 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; -2, 5, and 3+2i are zeros; f(1) = -96

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x^5−x^4−7x^3+7x^2−12x−12

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=11x^5-x^3+7x-5

Find the average rate of change of f(x)=√x from x1=4 to x2=9.

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. x^3−2x^2−11x+12=0

In Exercises 39–52, 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)=x^4−2x^3+x^2+12x+8

Finding the Equation of a Polynomial from a Graph

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)=4x^3−8x^2−3x+9

Rational Roots Test / Theorem

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=9x^6-7x^4+8x^2+x+6

Understanding Polynomial Functions

Finding all the Zeros of a Polynomial - Example 3

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. 2x^4+5x^3-2x^2+5x+6; x+3

Factor ƒ(x) into linear factors given that k is a zero. See Example 2. ƒ(x)=x^4+2x^3-7x^2-20x-12; k=-2 (multiplicity 2)

In Exercises 16–17, 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(x) = -2(x - 1)(x + 2)^2(x+5)^2

In Exercises 39–52, 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. x^4−3x^3−20x^2−24x−8=0

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. See Examples 4–6. 
2-i and 6-3i

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. 2x^3+x+2; x+1

In Exercises 39–52, 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)=x^3−4x^2−7x+10

In Exercises 39–52, 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. 2x^5+7x^4−18x^2−8x+8=0

In Exercises 25–32, 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?

Pre-Calculus - Using Descartes rule of signs

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=4x^4−x^3+5x^2−2x−6

Determine whether each statement is true or false. If false, explain why. Because x-1 is a factor of ƒ(x)=x^6-x^4+2x^2-2, 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 ƒ(x)=x^3+3x^2-4x-2.

Ex: The Rational Root (Zero) Theorem

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.

In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=5x^3−3x^2+3x−1

In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=x^3+2x^2+5x+4

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=5x^4+3x^2+2x-9

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=-8x^4+3x^3-6x^2+5x-7

In Exercises 39–52, 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. 4x^4−x^3+5x^2−2x−6=0

Finding all the Zeros of a Polynomial - Example 1

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. See Example 7. ƒ(x)=2x^5-7x^3+6x+8

For each polynomial function, one zero is given. Find all other zeros. See Examples 2 and 6. ƒ(x)=x^3+4x^2-5; 1

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x^2+4x+6=0

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. See Example 4. 
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. See Example 7. ƒ(x)=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. See Example 7. ƒ(x)=-2x^5+10x^4-6x^3+8x^2-x+1

Solving Higher Degree Polynomials by Synthetic Division and the Rational Roots Test

How To Find The Real & Imaginary Solutions of Polynomial Equations

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=4x^3+3x^2+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)=x^3−2x^2−11x+12

Determine the equation of a polynomial by analyzing its graph

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)=x^3+4x^2−3x−6

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. g(x) = x^4 - 6x^3 + x^2 + 24x + 16

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=3x^4+2x^3-8x^2-10x-1

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=3x^3+6x^2+x+7

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1. <p><img src="https://lightcat-files.s3.amazonaws.com/problem_images/dda4f49dc11fe8eb-1672428750420.jpg" alt="" /></p>

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=2x^3-4x^2+2x+7

Finding the Zeros of a Polynomial from Start to Finish

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)=4x^3-37x^2+50x+60

Find the zero in part (b) to three decimal places.

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(2x^2-7x+3)^3(x-2-√5)

Factor ƒ(x) into linear factors given that k is a zero. See Example 2. ƒ(x)=-6x^3-25x^2-3x+4; k=-4

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)=2x^4−3x^3−7x^2−8x+6

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. 4x^2+2x+54; x-4

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=4x^3-x^2+2x-7

Find the Zeros of a Polynomial Function with Imaginary Zeros

Rational Root Theorem

Find all zeros of f(x) = x³ + 5x² – 8x + 2.

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x^4−11x^3−3x^2−6x+8

Factor ƒ(x) into linear factors given that k is a zero. See Example 2. ƒ(x)=2x^4+x^3-9x^2-13x-5; k=-1 (multiplicity 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. x^3−10x−12=0

Finding All Zeros of a Polynomial Equation

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=6x^4+2x^3+9x^2+x+5

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

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x^2+4x−1=0

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^5-6x^4+14x^3-20x^2+24x-16

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. 5x^4+16x^3-15x^2+8x+16; x+4

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x^3+x^2−4x−4

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=2x^4-x^3+7x^2-4x-4

Determine whether each statement is true or false. If false, explain why. The polynomial function ƒ(x)=2x^5+3x^4-8x^3-5x+6 has three variations in sign.

Determine whether each statement is true or false. If false, explain why. For ƒ(x)=(x+2)^4(x-3), the number 2 is a zero of multiplicity 4.

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. x^3+2x^2+3; x-1

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. See Example 7. ƒ(x)=x^3+2x^2+x-10

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

In Exercises 37–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x) = 3x^4 - 2x^3 - 8x + 5

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