Question

in exercises 1-14, find the quotient and remainder when the first polynomial is divided by the second. you may use synthetic division wherever applicable.

1. (2x^2 + 13x + 15; x + 5)
2. (2x^2 - 7x + 3; x - 3)
3. (2x^3 - x^2 - 8x + 4; 2x - 1)
4. (3x^3 + 2x^2 - 3x - 2; 3x + 2)
5. (x^3 - 3x^2 + 2x - 4; x + 2)
6. (x^3 + 2x^2 - x - 3; x - 3)
7. (-3x^4 + x^2 - 2; 3x - 1)
8. (2x^4 - x^3 + x^2 - x; 2x + 1)
9. (x^6 + 1; x + 1)
10. (-x^3 + x; x - 5)
11. (x^3 + 2x^2 - 5; x^2 - 2)
12. (-x^3 - 3x^2 + 6; x^2 + 1)
13. (x^5 - x^4 + 2x^3 + x^2 - x + 1; x^3 + x - 1)
14. (-2x^5 + x^4 - x^3 + 2x^2 - 1; x^3 + x^2 + 1)

in exercises 15-20, write each polynomial in the form (p(x) = d(x)q(x) + r(x)). you may use synthetic division wherever applicable.

1. (x^2 + x + 1; d(x) = x + 1)
2. (x^2 + x + 1; d(x) = x - 1)
3. (3x^3 + 2x - 8; d(x) = x - 4)
4. (4x^3 - x + 4; d(x) = x - 2)
5. (x^6 - 3x^5 + x^4 - 2x^2 - 5x + 6; d(x) = x^2 + 2)
6. (-x^6 + 4x^5 - x^3 + x^2 + x - 8; d(x) = x^2 + 4)

in exercises 21-28, use synthetic division to find the function values.

1. (f(x) = x^3 - 7x + 5); find (f(3)) and (f(5)).
2. (f(x) = -2x^3 + 4x^2 - 7); find (f(4)) and (f(-3)).
3. (f(x) = -2x^4 - 10x^3 - 3x + 10); find (f(-1)) and (f(2)).
4. (f(x) = -x^4 + 3x^3 - 2x - 4); find (f(-2)) and (f(3)).
5. (f(x) = x^5 - 2x^3 + 12); find (f(3)) and (f(-2)).
6. (f(x) = -2x^5 + x^4 + x^2 - 2); find (f(-3)) and (f(4)).
7. (f(x) = x^4 - 2x^2 + 1); find (fleft(\frac{1}{2}

ight)).

1. (f(x) = -x^4 + 3x^2 - 2x); find (fleft(\frac{3}{2}

ight)).

in exercises 29-38, determine whether (q(x)) is a factor of (p(x)) and justify your answer.

1. (p(x) = x^3 - 7x + 6; q(x) = x - 3)
2. (p(x) = x^3 - 5x^2 + 8x - 4; q(x) = x + 2)
3. (p(x) = x^3 - 7x + 6; q(x) = x + 3)
4. (p(x) = x^3 - 5x^2 + 8x - 4; q(x) = x - 2)

Explanation:

Identify the scope of the request

The user has provided an image containing a large set of exercises (Exercises 1–38) on polynomial division, synthetic division, the Remainder Theorem, and the Factor Theorem. Since the user did not specify a single problem, we will solve the very first representative problem, Exercise 1, to demonstrate the method clearly.

Exercise 1: Find the quotient and remainder when the first polynomial is divided by the second:

$$(2x^2 + 13x + 15) \div (x + 5)$$

Set up synthetic division

Using the Synthetic Division knowledge point, we set up the division.
The divisor is \(x + 5\), so the test value is \(c = -5\).
The coefficients of the dividend \(2x^2 + 13x + 15\) are \(2\), \(13\), and \(15\).

Perform synthetic division calculations

Using the Synthetic Division knowledge point:

1. Bring down the leading coefficient: \(2\).
2. Multiply by \(-5\): \(-10\). Add to \(13\): \(13 + (-10) = 3\).
3. Multiply by \(-5\): \(-15\). Add to \(15\): \(15 + (-15) = 0\).

The resulting coefficients are \(2\) and \(3\), with a remainder of \(0\).

Interpret the results

The quotient is \(2x + 3\) and the remainder is \(0\).

Answer:

For Exercise 1:
Quotient: \(2x + 3\)
Remainder: \(0\)