Could you please solve so I can double check my answers for the practice quiz? Thank You!!
1. Solve x^3 + 6x^2 + 13x + 10 = 0.
a) –2 + 2i, –2 –2i, –2
b) 2 + i, 2 – i, –2
c) –2 + i, –2 – i, –2
d) 2 + 2i, 2 – 2i, –2
2. Find the discriminant and describe the nature of the roots of 3x^2 + 6x + 3 = 0.
a) The discriminant is 0 and there is/are one real root.
b) The discriminant is 4 and there is/are one real root.
c) The discriminant is 0 and there is no real root.
d) The discriminant is 7 and there is/are one real root.
3. Use the remainder theorem to find which of the following is not a factor of x^3 + 12x^2 + 47x + 60.
a) x + 5
b) x – 5
c) x + 4
d) x + 3
4. List all the rational roots of f(x) = 24x^3 – 10x^2 – 7x + 2.
a) -1/2, 1/4 ,2/3
b) -2/3,1/4 ,2/3
c) -1/2,1/3,1/2
d) -2/3,1/2,-1/4
5. Use the rational-root theorem to find the roots of 12x^3 – 8x^2 – 3x + 2 = 0.
a) -2/3,1/2,1/4
b) 2/3,-1/2,1/2
c) -1/2,1/2,1/4
d) 2/3,-1/3,1/3
what is your thinking? We don't do tests for you.
That would be cheating.
it is not a test, it is a practice quiz they helped us make in order to prepare for the test, I just want help double checking my answers
That's fine -- but I don't see YOUR answers. If you post them, someone will check them for you.
1) a
2) b
3) d
4) a
5) d
Were those wild guesses? In #2, the discriminant (b^2 = 4ac) is obviously not 4.
Sure! Let's go through each question step by step.
1. To solve the equation x^3 + 6x^2 + 13x + 10 = 0, we can use either factoring, the rational root theorem, or synthetic division. Let's use the rational root theorem to find the possible rational roots of the equation.
The rational root theorem states that if a polynomial equation has a rational root in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then it can be a potential root of the equation.
In this case, the constant term is 10 and the leading coefficient is 1. So the factors of 10 are ±1, ±2, ±5, and ±10, and the factors of 1 (leading coefficient) are ±1. Therefore, the potential rational roots are ±1, ±2, ±5, and ±10.
By trying out these potential roots and plugging them into the equation, we find that -2 is a root. Therefore, the correct answer is option c) –2 + i, –2 – i, –2.
2. The discriminant of a quadratic equation of the form ax^2 + bx + c = 0 is given by b^2 - 4ac. In the equation 3x^2 + 6x + 3 = 0, a = 3, b = 6, and c = 3. Plugging these values into the discriminant formula, we get 6^2 - 4(3)(3) = 36 - 36 = 0.
When the discriminant is equal to 0, the quadratic equation has one real root. So the correct answer is option a) The discriminant is 0 and there is/are one real root.
3. The remainder theorem states that if a polynomial f(x) is divided by (x - r), then the remainder will be f(r). To find which of the options is not a factor of x^3 + 12x^2 + 47x + 60, we can use the remainder theorem and try out each option as the divisor.
If we divide x^3 + 12x^2 + 47x + 60 by (x + 5), we should get a remainder of 0. Let's perform this division:
Using synthetic division:
-5 | 1 12 47 60
| -5 -35 -60
----------------------
1 7 12 0
The remainder is indeed 0, so x + 5 is a factor.
Now let's try dividing by the other options and see which one does not result in a remainder of 0.
By trying out each option as a divisor using synthetic division, we find that x - 5 does not result in a remainder of 0. Therefore, the correct answer is option b) x - 5.
4. To find the rational roots of the polynomial f(x) = 24x^3 – 10x^2 – 7x + 2, we can use the rational root theorem. The potential rational roots are the factors of the constant term 2 (±1, ±2) divided by the factors of the leading coefficient 24 (±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24).
By trying out each potential rational root and using synthetic division, we find that -1/2 and 2/3 are the rational roots of the polynomial. Therefore, the correct answer is option a) -1/2, 1/4, 2/3.
5. To determine the roots of the equation 12x^3 – 8x^2 – 3x + 2 = 0 using the rational root theorem, we need to find the potential rational roots. The factors of the constant term 2 are ±1, ±2, and the factors of the leading coefficient 12 are ±1, ±2, ±3, ±4, ±6, ±12.
After trying out these potential rational roots using synthetic division, we find that the roots of the equation are -2/3 and 1/2. Therefore, the correct answer is option b) -2/3, 1/2, 1/2.
I hope this helps you double-check your answers! Let me know if you have any further questions.