College Algebra
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Solve the given equation. a4 + a3 - 7a2 - 5a + 10 = 0
Identify the nth-degree polynomial function with real coefficients that satisfies the following conditions: n = 3, -2 and 2 + 5i are zeros, f(1) = 78.
Identify the nth-degree polynomial function with real coefficients that satisfies the following conditions: n = 3, 2 and 4i are zeros, f(1) = -34.
List the possible rational zeros of the following equation using Rational Zero Theorem. Then, use synthetic division to find an actual zero. Find the remaining zeros using the quotient from division: x3 - 3x2 - 5x - 1 = 0
Find x.
x4 + 8x3 + 11x2 - 40x - 80 = 0
Find the zeros of the following function: f(x) = x3 + x2 - 15x + 9
Consider the given function. Calculate its average rate of change from x1 = 16 to x2 = 25.
f(x) = 2√x
Find the zeros of the given polynomial function. Use Descartes's Rule of Signs and Rational Zero Theorem. You may also use a graphing utility to help you get the first zero. f(x) = 2x4 - 5x3 - 15x2 + 10x + 8
Find the zeros of the given polynomial function. Use Descartes's Rule of Signs and Rational Zero Theorem. You may also use a graphing utility to help you get the first zero.
f(x) = x3 - x2 - 14x + 24
If f(x) = an(2x4 - 5x2 - 1) and f(2) = 132, determine the value of an?
List the possible rational zeros of the following function using Rational Zero Theorem. Then, use synthetic division to find an actual zero. Find the remaining zeros using the quotient from division: f(x) = 2x3 + x2 - 41x + 20
List the possible rational zeros of the following function using Rational Zero Theorem. Then, use synthetic division to find an actual zero. Find the remaining zeros using the quotient from division: f(x) = x3 - 6x2 + 3x + 10
Solve the following equation: x2 + 6x + 11 = 0
Solve the following equation: x2 + 6x + 2 = 0
Identify the possible number of positive and negative real zeros of the following function by using the Descartes's Rule of Signs:
f(x) = 6x4 - x3 - 4x2 - 2x + 9
f(x) = 8x3 - 6x2 + 9x - 3
f(x) = 3x3 + 4x2 + 7x + 1
List the possible rational zeros of the following function using the Rational Zero Theorem:
f(x) = 3x5 - 5x4 - 2x3 +9x2 - 15x - 16
f(x) = 5x4 - 7x3 - 2x2 - 9x + 6
f(x) = 6x4 - 3x3 + 8x2 - 5x - 9
f(x) = 7x4 - 8x3 - 2x2 + 16x + 9
f(x) = x4 - x2 + 5x - 8
Specify the number of real zeros and the number of imaginary zeros using the graph of a fifth-degree polynomial shown below.
Consider the conditions of an unknown polynomial: n = 3; 1 and (- 1 + 4i) are zeros; f(2) = 50. Find this nth-degree polynomial.
Express this polynomial as a product of its linear and quadratic factors. To do that, first, you need to find all zeros of this polynomial. x4 - 2x3 - 3x2 + 4x + 4
Express this polynomial as a product of its linear and quadratic factors. To do that, first you need to find all zeros of this polynomial.
3x4 + x3 + 4x2 + 2x - 4
For the given polynomial function, use Descartes' Rule of signs to find the possible rational zeros and actual rational zeros.
f(x) = 2x3 - 4x2 - 1
For the given polynomial function, find the zeros and for each zero, state its multiplicity and comment if it crosses the x-axis or touches the x-axis and turns around.
f(x) = - 4(x + 1)(x - 3)3(x - 2)4
Consider the following polynomial: 9x4 + 3x2 + 6.
Explain why it has no real roots. Hint: Use Descartes' Rule of Signs.
Find the possible number of positive and negative real zeros for the given function by using Descartes' Rule of Signs.
f(x) = 2x4 + 3x3 - 4x + 1
Find all possible rational zeros for the given function 4x4 - 2x3 + 10x2 - 8x + 1 using the Rational Zero Theorem.
Consider the following polynomial: f(x) = x3 - 5x + 1.
Prove that it has a real zero between 0 and 1.
If 8 + 9i is the one of the zero of the function f(x) having real coefficients, write the other zero of the function.
Factor the following function to express it as a product of its linear factors if 4 is a zero. f(x) = 4x3 - 35x2 + 71x + 20
Factor the following function to express it as a product of its linear factors if - 1 is a zero. f(x) = - 20x3 - 18x2 + 6x + 4
Factor the following function to express it as a product of its linear factors if - 3 (multiplicity 2) is a zero. f(x) = x4 + 5x3 - 17x2 - 129x - 180
Factor the following function to express it as a product of its linear factors if - 6 (multiplicity 3) is a zero. f(x) = x4 + 17x3 + 90x2 + 108x - 216
If 5 is a zero of the following polynomial function, solve for all the other zeros. f(x) = x3 - 9x2 + 25x - 25
If 1 is a zero of the following polynomial function, solve for all the other zeros. f(x) = x3 + 13x2 - 14.
If 1/3 is a zero of the following polynomial function, solve for all the other zeros. f(x) = 3x3 - 43x2 + 56x - 14
If 3i is a zero of the following polynomial function, solve for all the other zeros. f(x) = x4 + 25x2 + 144.
Identify whether the given statement is true or not. If it is false, then write the reason.
If x - 2 is a factor of f(x) = x6 - 2x4 + 7x2 - 60, then f(2) = 0.
5 is a zero of the function ƒ(x) = (x + 5)5(x - 7) with multiplicity of 5.
A polynomial function of degree 4 and having real coefficients may have no real zeros.
There are four sign variations in the function ƒ(x) = 4x5 - 7x4 + 8x3 + 5x - 6.
The multiplication of the complex number with its conjugate always gives a real number.
For the following polynomial, perform factor theorem and synthetic division to identify whether x - 3 is a factor or not.
f(x) = x3 + 2x2 - 5x - 6
For the following polynomial, perform factor theorem and synthetic division to identify whether x + 2 is a factor or not.
f(x) = x3 + 4x2 - 11x - 30
For the following polynomial, perform factor theorem and synthetic division to identify whether x - 5 is a factor or not.
f(x) = 5x2 + 9x + 60
For the following polynomial, perform factor theorem and synthetic division to identify whether x + 3 is a factor or not.
f(x) = 9x3 + 11x + 50
For the following polynomial, perform factor theorem and synthetic division to identify whether x + 9 is a factor or not.
f(x) = 6x4 + 35x3 - 317x2 - 1419x - 945
Identify the zeros and their multiplicities for the following function. f(x) = (x + 7)3(x - 7)2(x2 - 45)
Identify the zeros and their multiplicities for the following function. f(x) = 7x(x - 9)(x + 4)(x2 - 25)
Identify the zeros and their multiplicities for the following function. f(x) = 7x3(x2 - 36)(x + 4)
Identify the zeros and their multiplicities for the following function. f(x) = (x2 + 2x - 24)7(x - 2 + √5)4
Provide the polynomial function equation with zeros -5, 2, and 3 and f(4) = 54. The degree of the polynomial function is 3.
Provide the polynomial function with zero at -5 having multiplicity 3 and f(-4) = 7. The degree of the polynomial function is 3.
Using the zeros 11 +2i and 11 -2i, write the equation of a polynomial function of the least degree. The coefficient of the polynomial must be only real.
Using the zeros 6 -i, 4, and -3, write the equation of a polynomial function of the least degree. The coefficient of the polynomial must be only real and take multiplicity 1 unless otherwise stated.
Using the zeros 10 -i and 8 -4i, write the equation of a polynomial function of the least degree. The coefficient of the polynomial must be only real and take multiplicity 1 unless otherwise stated.
Solve for a certain polynomial function having the following zeros if it were to have the least degree and real coefficients. - √5, √5, 4, 5
Solve for a certain polynomial function having the following zeros if it were to have the least degree and real coefficients. - 3 + √6, - 3 - √6, - 5, 4
For the following polynomial function, solve for its rational roots. f(x) = 6x4 - 19x3 - 37x2 + 62x + 24
For the following polynomial function, prove that it has a real zero that lies between - 1 and 0.
f(x) = 4x3 - 21x2 + 16x + 5
For the following polynomial function, prove that it has a real zero that lies between 1 and 2.
For the following polynomial function, evaluate the zero that lies between 1 and 2 and write in two decimal places.
For the following polynomial function, prove that it has a real zero that lies between 11 and 12.
f(x) = 7x3 - 127x2 + 536x - 74
For the following polynomial function, prove that it has a real zero that lies between 6 and 7.
For the following polynomial function, evaluate the zero that lies between 6 and 7 and write in four decimal places.
For the following function, identify the possible number of positive, negative and nonreal complex zeros using Descartes' Rule of Signs.
f(x) = 5x3 + 9x2 - 8x - 7
Prove that x + 3 is a factor of the polynomial function f(x) = 9x3 + 33x2 + 29x + 33.
For the polynomial function f(x) = 13x3 - 7x2 + 69, determine the maximum number of turning points of its graph.
For the following polynomial function, determine all the complex zeros. Write the answer in exact form.
f(x) = x4 + 3x3 - 20x2 + 24x - 224
f(x) = x4 - 3x3 - 6x2 + 28x - 24
f(x) = 3x5 + 4x4 + 7x3 + 14x2 + 8x
f(x) = x5 - 3x4 + 9x3 - 19x2 + 18x - 6
f(x) = 3x3 - 10x2 + 22x + 20
f(x) = x4 + 8x3 + 24x2 + 32x + 16
f(x) = x4 + 4x2 + 4
f(x) = x4 - 8x3 + 14x2
f(x) = x4 - 8x3 + 26x2 - 40x + 24
f(x) = x6 - 25x4 - 81x2 + 2025
For the following polynomial function, prove that it has no zero greater than 2.
f(x) = 9x4 + 5x3 - 6x2 - 15x + 8
For the following polynomial function, prove that it has no zero less than - 2.
Identify the polynomial function given the following features:
Degree: 3Zeros: - 5, 2 and 6f(4) = 72
For the given function, find the various potential scenarios for the count of positive, negative, and nonreal complex zeros.
ƒ(x) = 9x3 -13x2 +7x +1
ƒ(x) = 12x3 +3x2 +71x -110
ƒ(x) = 8x3 -7x2 +15x -13
For the numbers of positive, negative, and nonreal complex zeros, find out all possibilities.
ƒ(x) = 14x3 +22x2 +3x +53
ƒ(x) = 4x4 +5x2 +5x -3
ƒ(x) = 6x4 +101x3 -17x2 -3x -10
ƒ(x) = -15x4 +x3 -12x2 +5x -16
ƒ(x) = 15x4 +3x3 +25x2 +3x +1
ƒ(x) = x5 + 11x4 - 6x3 + 19x + 7
ƒ(x) = 3x5 -34x4 +57x3 -11x2 +2x +9
ƒ(x) = 4x5 - 7x3 + 9x + 15
ƒ(x) = 9x5 -31x3 +5x +7
ƒ(x) = 2x6 -16x5 +27x3 -5x2 +11x +1
ƒ(x) = 13x5 +8x4 +7x3 +17x2 +5x +15
ƒ(x) = -13x5 +5x4 -6x3 +4x2 -9x +36
List the possible rational zeros of the following function using Rational Zero Theorem. Then, use synthetic division to find an actual zero. Find the remaining zeros using the quotient from division: f(x) = x3 + 3x2 - 10x - 24