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### Problem 8: Polynomial Multiplication and Remainder

We need to determine the remainder \( r \) to make each multiplication statement true. Let's look at the first one:

**a) \((2x - 3)(3x + 5) + r = 6x^2 + x + 5\)**

1. Expand \((2x - 3)(3x + 5)\):
   \[
   (2x - 3)(3x + 5) = 2x \cdot 3x + 2x \cdot 5 - 3 \cdot 3x - 3 \cdot 5
   \]
   \[
   = 6x^2 + 10x - 9x - 15 = 6x^2 + x - 15
   \]

2. Now, we compare this result with \( 6x^2 + x + 5 \). To make the equation equal, the remainder \( r \) must make up for the difference between the constants \(-15\) and \(+5\). Thus,
   \[
   r = 20.
   \]
So the answer for part (a) is \( r = 20 \).

Now, let's move on to part (b).

**b) \((x + 3)(x + 5) + r = x^2 + 9x - 7\)**

1. Expand \((x + 3)(x + 5)\):
   \[
   (x + 3)(x + 5) = x \cdot x + x \cdot 5 + 3 \cdot x + 3 \cdot 5
   \]
   \[
   = x^2 + 5x + 3x + 15 = x^2 + 8x + 15
   \]

2. Now, comparing this result with \( x^2 + 9x - 7 \), we see that:
   - To match the \(x\)-terms, the remainder \(r\) must include an extra \(x\) term: \(9x - 8x = x\).
   - To adjust the constant term, we need \( -7 - 15 = -22 \).

   Thus, \( r = x - 22 \).

For the remaining parts (c) and (d), the same approach would be followed. Would you like me to continue with those, or focus on Problem 9?

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### Problem 9: Polynomial Division and Finding the Divisor

In this problem, each dividend was divided by another polynomial, and we need to find the divisor using the quotient and remainder. Let's solve part (a):

**a) Dividend: \(5x^3 + x^2 + 3\), Quotient: \(5x^2 - 14x + 42\), Remainder: \(-123\)**

1. We know that the division follows:
   \[
   \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}
   \]
   Substituting what we know:
   \[
   5x^3 + x^2 + 3 = (\text{Divisor} \times (5x^2 - 14x + 42)) - 123
   \]
2. Solve this equation for the divisor by comparing the polynomial degrees.

Would you like me to go through the steps of polynomial long division here, or should I continue with the other parts?

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Let me know if you'd like more details on these or have any questions!

### Related Questions:
1. How do you perform polynomial long division step-by-step?
2. What is the remainder theorem and how does it apply here?
3. How do you verify if a polynomial division result is correct?
4. Can a remainder ever be larger than the divisor in polynomial division?
5. How does synthetic division differ from polynomial long division?

**Tip**: When dividing polynomials, always check that the degree of the remainder is less than the degree of the divisor to ensure the division is complete.

Determine the remainder to make the polynomial multiplication statements true and find the divisor in polynomial division using the given quotient and remainder.

Explore polynomial multiplication and division with detailed examples. Learn how to determine remainders and find divisors using quotients and remainders. Suitable for high school students in Grades 9-12.

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