I have three questions:
1. What is the remainder of (x^3 + 2x^2 - 5x+4) /(x-3) I think the answer is 34??
2. Evaluate (4+5i)(2+2i)
=4(2) +4(2i) + 5i(2) + 5i(2i) = 8 + 8i + 10i+10i^2(-1)
=-2 + 18i
3. Using Rational Roots Theorem, list all possible rational root of f(x) = 2x^3 - 3x^2 + 5x+6
Possible roots are p/a = -+1, -+2, +-3, -+6. -+3/2
1. is correct
2. is correct
Number three should read p/q = not p/a
Thank you for letting me know the first two are correct
1. To find the remainder of the polynomial (x^3 + 2x^2 - 5x + 4) divided by (x - 3), you can use polynomial long division or synthetic division.
Let's use synthetic division to solve this:
Set up the synthetic division problem like this:
3 | 1 2 -5 4
Start by bringing down the 1 (coefficient of x^3):
3 | 1 2 -5 4
---------------------
1
Next, multiply the divisor (3) by the quotient (which is 1 in this case) and subtract the result from the next coefficient:
3 | 1 2 -5 4
---------------------
1
-3
---
-1
Repeat the process by bringing down the next coefficient (-5) and continue:
3 | 1 2 -5 4
---------------------
1
-3
---
-1 -3
-----
-6
Finally, bring down the last coefficient (4), and the resulting divisor is the remainder:
3 | 1 2 -5 4
---------------------
1
-3
---
-1 -3
-----
-6 1
So the remainder is 1, not 34.
2. To evaluate the expression (4 + 5i)(2 + 2i), you can use the FOIL method.
FOIL stands for: First, Outer, Inner, and Last.
First: Multiply the first terms: 4 * 2 = 8
Outer: Multiply the outer terms: 4 * 2i = 8i
Inner: Multiply the inner terms: 5i * 2 = 10i
Last: Multiply the last terms: 5i * 2i = 10i^2 = -10
Combine all the terms:
8 + 8i + 10i - 10
Simplify:
8 + 18i - 10
Result:
-2 + 18i
So the result is -2 + 18i, not -2 - 18i.
3. To find the possible rational roots of the polynomial f(x) = 2x^3 - 3x^2 + 5x + 6 using the Rational Roots Theorem, you need to consider the factors of the constant term (which is 6) divided by the factors of the leading coefficient (which is 2).
The factors of 6 are: ±1, ±2, ±3, and ±6.
And the factors of 2 are: ±1 and ±2.
So the possible rational roots, expressed as p/a, where p is a factor of the constant term and a is a factor of the leading coefficient, are:
±1/1, ±2/1, ±3/1, ±6/1, ±1/2, and ±3/2.
Therefore, the possible rational roots of f(x) = 2x^3 - 3x^2 + 5x + 6 are ±1, ±2, ±3, ±6, ±1/2, and ±3/2.