7. Divide -2x^3 - 5x^2 + 4x + 2 by x + 2
A. -2x^2-x+6
B. -2x^2-x+6, R 14
C. -2x^2-x+6, R 10
D. -2x^2-9x+22, R -42
8. x^3 - 4x^2 + x + 26 = 0
A. 3+_ 2i, -2 *I think its this one*
B. -3+_2i, -2
C. 3+_2i, 2
D. -3+_2i, 2
#7 I like C, but with R = -10
#8 correct
Oh yeah it is actually -10 I made a typo, thank you.
7. Well, dividing can be quite tricky, just like dividing a pizza between squabbling siblings. But fear not, my friend, I am here to lend a hand, or rather, a funny answer! Let's divide -2x^3 - 5x^2 + 4x + 2 by x + 2. The answer is:
B. -2x^2 - x + 6, with a remainder of 14. Because hey, sometimes you just have to have some leftover pizza!
8. Ah, solving cubic equations, the ultimate challenge! But fear not, I'll give you an answer that'll make you smile. The solutions to x^3 - 4x^2 + x + 26 = 0 are:
D. -3 ± 2i, 2. Keep in mind that "i" represents the imaginary unit, just like the imaginary friends we had when we were kids. So, grab your imaginary friend, and say hello to the answer!
To divide -2x^3 - 5x^2 + 4x + 2 by x + 2, you can use polynomial long division. Here are the steps:
Step 1: Divide the first term of the dividend by the first term of the divisor.
-2x^3 divided by x gives -2x^2.
Step 2: Multiply the divisor (x + 2) by the quotient from Step 1 (-2x^2).
(x + 2) * (-2x^2) = -2x^3 - 4x^2.
Step 3: Subtract the result from Step 2 from the dividend.
(-2x^3 - 5x^2 + 4x + 2) - (-2x^3 - 4x^2) = -x^2 + 4x + 2.
Step 4: Bring down the next term from the dividend.
You bring down the 4x: -x^2 + 4x + 2.
Step 5: Divide the first term of the new dividend by the first term of the divisor.
(-x^2) divided by x gives -x.
Step 6: Multiply the divisor (x + 2) by the quotient from Step 5 (-x).
(x + 2) * (-x) = -x^2 - 2x.
Step 7: Subtract the result from Step 6 from the new dividend.
(-x^2 + 4x + 2) - (-x^2 - 2x) = 6x + 2.
Step 8: Bring down the next term from the dividend.
You bring down the 6x: 6x + 2.
Step 9: Divide the first term of the new dividend by the first term of the divisor.
(6x) divided by x gives 6.
Step 10: Multiply the divisor (x + 2) by the quotient from Step 9 (6).
(x + 2) * 6 = 6x + 12.
Step 11: Subtract the result from Step 10 from the new dividend.
(6x + 2) - (6x + 12) = -10.
At this point, there is no more term to bring down, and the remainder is -10.
So, the final answer is -2x^2 - x + 6 with a remainder of -10. Therefore, the correct answer is (C) -2x^2 - x + 6, R 10.
Now, moving on to question 8, to find the roots of the equation x^3 - 4x^2 + x + 26 = 0, you can use either factoring or synthetic division. Since the equation is not factorable, we will use synthetic division.
Using synthetic division with the possible root -2, we have:
-2 | 1 -4 1 26
| -2 12 -26
__________________
1 -6 13 0
The result is: x^2 - 6x + 13 = 0.
To find the roots of this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a.
For the equation x^2 - 6x + 13 = 0, a = 1, b = -6, and c = 13.
Plugging in the values, we get:
x = (-(-6) ± √((-6)^2 - 4(1)(13))) / (2(1))
= (6 ± √(36 - 52)) / 2
= (6 ± √(-16)) / 2
= (6 ± 4i) / 2
= 3 ± 2i.
Therefore, the correct answer is (A) 3 ± 2i, -2.
To divide a polynomial by another polynomial, we use polynomial long division. Here is how you can solve the first problem step by step:
1. Write the dividend (the polynomial being divided) and divisor (the polynomial we're dividing by) in descending order of powers of x:
Dividend: -2x^3 - 5x^2 + 4x + 2
Divisor: x + 2
2. Divide the first term of the dividend (-2x^3) by the first term of the divisor (x). The result is -2x^2.
Write this result above the long division symbol.
3. Multiply the divisor (x + 2) by the result (-2x^2). The result is -2x^3 - 4x^2.
Write this result below the dividend, aligning the like terms.
-2x^2
_____________________
x + 2 | -2x^3 - 5x^2 + 4x + 2
-2x^3 - 4x^2
______________
4. Subtract the latest multiplication result from the dividend. -2x^3 - 5x^2 (dividend) - (-2x^3 - 4x^2) (multiplication result) = -5x^2 + 4x^2 = -x^2.
Write this result below the line.
-2x^2
_____________________
x + 2 | -2x^3 - 5x^2 + 4x + 2
-2x^3 - 4x^2
______________
-x^2
5. Bring down the next term from the dividend, which is 4x. The expression now becomes -x^2 + 4x.
Write this below the line, aligned with the previous term.
-2x^2
_____________________
x + 2 | -2x^3 - 5x^2 + 4x + 2
-2x^3 - 4x^2
______________
-x^2 + 4x
6. Divide the first term of the new expression (-x^2) by the first term of the divisor (x). The result is -x.
Write this above the line, to the right of the previously obtained result.
-2x^2 - x
_____________________
x + 2 | -2x^3 - 5x^2 + 4x + 2
-2x^3 - 4x^2
______________
-x^2 + 4x
7. Multiply the divisor (x + 2) by the result obtained (-x). The result is -x^2 - 2x.
Write this result below the line, aligning like terms.
-2x^2 - x
_____________________
x + 2 | -2x^3 - 5x^2 + 4x + 2
-2x^3 - 4x^2
______________
-x^2 + 4x
-x^2 - 2x
______________
8. Subtract the latest multiplication result from the current expression. (-x^2 + 4x) - (-x^2 - 2x) = 6x.
Write this result below the line.
-2x^2 - x
_____________________
x + 2 | -2x^3 - 5x^2 + 4x + 2
-2x^3 - 4x^2
______________
-x^2 + 4x
-x^2 - 2x
______________
6x
9. Now, we have the expression 6x. There are no more terms to bring down from the dividend.
Since the degree of the divisor (x + 2) is 1, and the degree of the remaining expression (6x) is 1 as well,
we cannot divide further, and this is the remainder.
10. Write the final quotient and remainder. The quotient is -2x^2 - x and the remainder is 6x.
Combine them to form the final answer.
Therefore, the answer is A. -2x^2 - x + 6.
Now moving on to the second problem:
To solve the equation x^3 - 4x^2 + x + 26 = 0, we can use the rational root theorem and synthetic division.
1. List all the possible rational roots of the polynomial. These are the factors of the constant term (26) divided by the factors of the leading coefficient (1).
In this case, the factors of 26 are ±1, ±2, ±13, and ±26, and the factors of 1 are ±1.
So, the possible rational roots are ±1, ±2, ±13, and ±26.
2. Use synthetic division to find the actual roots. Start by trying a potential root and see if it gives a remainder of 0.
Let's try x = -2:
-2 | 1 -4 1 26
-2 12 -26
___________
1 -6 13 0
The remainder is zero, which means x = -2 is a root of the equation.
3. Use the quotient obtained from synthetic division to find the remaining roots.
The quotient obtained is 1 - 6x + 13x^2.
To find the remaining roots, solve 1 - 6x + 13x^2 = 0.
You can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (6 ± √((-6)^2 - 4(13)(1))) / (2(13))
x = (6 ± √(36 - 52)) / 26
x = (6 ± √(-16)) / 26
x = (6 ± 4i) / 26
Simplifying, we get:
x = (3 ± 2i) / 13
The remaining roots are (3 + 2i)/13 and (3 - 2i)/13.
4. Combine all the roots into the final answer.
Therefore, the answer is D. -3 ± 2i, 2.