1. Which equation is best represented by the graph?
a. y= (x+3)(x+1)(x-1)
b. y= (x-3)(x+1)(x-1)
c. y= -(x-3)(x+1)(x-1)
d. y= (3-x)(x+1)(x-1)
2. Shown below is the graph of y=x^3-3x^2-6x+8. What are the apparent zeros of the function graphed above?
a.{-0.7,-2.7)
b.{-10.2,10.2}
c.{-2,1,4}
d.{-4,-1,2}
3. What is a polynomial function in a standard form with zeros 1,2,3 and -3?
a. x^4-3x^3+7x^2-27-18
b. x^4+3x^3-7x^2-27-18
c. x^4-3x^3-7x^2+27-18
d. x^4+3x^3+7x^2-27-18
4. What are the zeros of the function? What are their multiplicities?
F(x)= 5x^3-5x^2-30x
a. The num.. 3,-2, and 0 are zeros of multiplicity 1.
b. The num.. 3,-2, and 0 are zeros of multiplicity 2.
c. The num..-3,2, and 0 are zeros of multiplicity 1.
d. The num..-3,2, and 0 are zeros of multiplicity 2.
5. What are the real and complex solutions of the polynomial equation?
x^3-64=0(The / in the equations below aren't fractions, just so you know!)
a. 4,-1+2i/3,-1+2i/3
b. 4,1+2i/3,1+2i/3
c. 4,-2+2i/3,-2+2i/3
d. 4,+2+2i/3,+2+2i/3
6. What is the equivalent to the following expression?
(3p^2+5pq-q^2)+(p^2+3pq-2q^2)
a. 3p^2+8pq-3q^2
b. 4p^2+8pq-q^2
c. 4p^4+8pq-3q^4
d. 4p^2+8pq-3q^2
7. Divide -3x^3-4x^2+4x+3 by x-2.
a. -3x^2+2x+24
b. -3x^2-10x-16, R-29
c. -3x^2-10x-16
d. -3x^2+2x+24, R 35
8. Find the roots of the polynomial equation. X^3-4x^2+x+26=0
a. 3(+-)2i,-2
b. -3(+-)2i,-2
c. 3(+-)2i, 2
d. -3(+-)2i, 2
9. Which correctly describes the roots of the following cubic equation?
x^3+6x^2+11x+6
a. One real rt., two complex rts.
b. Tow real rts. and one complex rt.
c. Three real rts. two of which are equal in value
d. Three real rts, each with a different value
10. One zero of f(x)=x^3-2x^2-5x+6 is 1. What are other zeros of the function?
a. 3 and -2
b. -3 and 2
c. 1 and 2
d. 1 and -3
What of the following quartic functions has x= -1 and x= -3 as its only two real zeros?
a. x^4-4x^3-4x^2-4x-3
b. x^4+4x^3+4x^2+4x+3
c. x^4+4x^3+3x^2+4x-4
d. x^4+4x^3+4x^2+4x+3
1. The best represented equation by the graph is option c. y= -(x-3)(x+1)(x-1). Because what's better than a function with a negative sign in front? It adds that extra "oomph" to the graph.
2. The apparent zeros of the function graphed above are option c. {-2, 1, 4}. These zeros are pretty apparent if you ask me, they're just hanging out there on the x-axis, not even trying to hide.
3. The polynomial function in standard form with zeros 1, 2, 3, and -3 is option b. x^4+3x^3-7x^2-27-18. This function has all the right zeros in all the right places, just like a perfectly coordinated dance routine.
4. The zeros of the function F(x)= 5x^3-5x^2-30x are 3, -2, and 0. But their multiplicities are option a. 1. That means these zeros are just one-time attendees at the party, not wanting to overstay their welcome.
5. The real and complex solutions of the polynomial equation x^3-64=0 are option b. 4, 1+2i/3, 1+2i/3. These solutions are as real as it gets, with a side dish of complex numbers to keep things interesting. It's like having both vanilla and chocolate ice cream at the same time.
6. The equivalent expression to (3p^2+5pq-q^2)+(p^2+3pq-2q^2) is option d. 4p^2+8pq-3q^2. This expression combines the best of both worlds, like a pizza topped with all your favorite toppings.
7. When you divide -3x^3-4x^2+4x+3 by x-2, you get option c. -3x^2-10x-16. It's a division miracle! And there's no remainder, so it's a clean break.
8. The roots of the polynomial equation X^3-4x^2+x+26=0 are option d. -3(+-)2i, 2. These roots are definitely out of this world, with a hint of realness thrown in for good measure.
9. The roots of the cubic equation x^3+6x^2+11x+6 are option c. Three real roots, two of which are equal in value. It's like having a double dose of realness, because why settle for just one root when you can have two?
10. If one zero of f(x)=x^3-2x^2-5x+6 is 1, then the other zeros of the function are option b. -3 and 2. They're like the perfect trio, hanging out together to solve the equation.
Finally, the quartic function that has x= -1 and x= -3 as its only two real zeros is option a. x^4-4x^3-4x^2-4x-3. It's like a magic trick, making those zeros disappear into the real number realm.
1. The equation best represented by the graph is d. y= (3-x)(x+1)(x-1).
2. The apparent zeros of the function graphed above are c. {-2, 1, 4}.
3. A polynomial function in standard form with zeros 1, 2, 3, and -3 is a. x^4-3x^3+7x^2-27x-18.
4. The zeros of the function F(x) = 5x^3-5x^2-30x are c. The numbers -3, 2, and 0 are zeros of multiplicity 1.
5. The real solution of the polynomial equation x^3-64=0 is a. 4. The complex solutions are b. -1+2i/3 and c. -1-2i/3.
6. The equivalent expression to (3p^2+5pq-q^2)+(p^2+3pq-2q^2) is b. 4p^2+8pq-q^2.
7. The division of -3x^3-4x^2+4x+3 by x-2 is c. -3x^2-10x-16.
8. The roots of the polynomial equation x^3-4x^2+x+26=0 are c. 3(+/-)2i and 2.
9. The roots of the cubic equation x^3+6x^2+11x+6 are a. One real root and two complex roots.
10. If one zero of f(x)=x^3-2x^2-5x+6 is 1, then the other zeros of the function are b. -3 and 2.
The quartic function that has x=-1 and x=-3 as its only two real zeros is c. x^4+4x^3+3x^2+4x-4.
1. To determine the equation represented by the graph, we need to analyze the behavior of the graph. From the graph, we can observe that the graph intersects the x-axis at x=-3, x=1, and x=-1.
a. y= (x+3)(x+1)(x-1) - This equation represents the graph because it has roots at x=-3, x=1, and x=-1, which match the points where the graph intersects the x-axis.
2. To find the apparent zeros of the function, we need to find the x-values where the graph intersects the x-axis.
b. {-10.2, 10.2} - The apparent zeros of the function are approximate values of -10.2 and 10.2, which do not match any points on the graph.
c. {-2, 1, 4} - The apparent zeros of the function are -2, 1, and 4, which do not match any points on the graph.
d. {-4, -1, 2} - The apparent zeros of the function are -4, -1, and 2, which do not match any points on the graph.
Therefore, none of the options accurately represent the apparent zeros of the graph.
3. To find a polynomial function with the given zeros, we can use the factored form of a polynomial equation.
b. x^4+3x^3-7x^2-27-18 - This equation does not accurately represent the zeros of the function.
c. x^4-3x^3-7x^2+27-18 - This equation accurately represents the zeros of the function as it has the factors (x-1)(x-2)(x-3)(x+3).
4. To find the zeros and their multiplicities, we need to factor the polynomial.
c. The numbers -3, 2, and 0 are zeros of multiplicity 1 - This statement accurately represents the zeros and their multiplicities. The given function can be factored as F(x) = 5x(x-3)(x+2).
5. To find the real and complex solutions of the polynomial equation, we need to solve the equation.
a. 4, -1+2i/3, -1+2i/3 - This option accurately represents the real and complex solutions of the polynomial equation x^3-64=0. The real solution is x = 4, and the complex solutions are x = -1 + 2i/3 and x = -1 - 2i/3.
6. To simplify the given expression, we need to combine like terms.
b. 4p^2+8pq-q^2 - This equation accurately represents the simplification of the expression (3p^2+5pq-q^2)+(p^2+3pq-2q^2).
7. To divide the given polynomial by x-2, we can use long division or synthetic division.
c. -3x^2-10x-16 - This equation accurately represents the division of -3x^3-4x^2+4x+3 by x-2.
8. To find the roots of the polynomial equation, we can use factoring, synthetic division, or a numerical method such as the Newton-Raphson method.
c. 3(+-)2i, 2 - This equation accurately represents the roots of the polynomial equation X^3-4x^2+x+26=0. The roots are x = 3+2i, x = 3-2i, and x = 2.
9. To determine the nature of the roots of the cubic equation, we need to analyze the discriminant and the behavior of the function.
d. Three real roots, each with a different value - This statement accurately describes the roots of the cubic equation x^3+6x^2+11x+6.
10. Since we are given one zero of the function, which is 1, we can use synthetic division, factoring, or a numerical method to find the remaining zeros.
a. 3 and -2 - This option accurately represents the other zeros of the function f(x)=x^3-2x^2-5x+6. The zeros are x = 1, x = 3, and x = -2.
11. To find a quartic function with specific real zeros, we can use the factored form of a polynomial equation.
b. x^4+4x^3+4x^2+4x+3 - This equation accurately represents a quartic function with x = -1 and x = -3 as its only real zeros.
sorry, no graphs on this web site.
interesting problems.
Did you try to do any of them?
What are your answer choices?
I'll do one to get you stated.
#9 x^3+6x^2+11x+6
by Descartes' Rule of Signs, there are 0 or 2 positive roots, and one or three negative root.
by the Rational Root Theorem, the negative root must be -1, -2, -3 or -6
A little checking show that x = -1 is one root, giving
(x+1)(x^2 + 5x + 6) = (x+1)(x+2)(x+3)