Find the least common multiple of the expressions below. Once found, determine the
quantities with which you need to multiply each expression in order to obtain the
least common multiple.
(a) x and x + 3.
(b) x2 + 6x − 16 and x2 + 9x + 14.
(c) x3 − 36x and x2 + 2x − 24.
(d) x2 + 2tx − 3t2 and x2 − t2.

### (a) \( x \) and \( x + 3 \)

- **LCM of \( x \) and \( x + 3 \):**
  Since these two expressions are linear and distinct, their least common multiple is simply their product.
  
  \[
  \text{LCM}(x, x + 3) = x(x + 3)
  \]

- **Multiplying factors:**
  - To make \( x \) into the LCM, you need to multiply \( x \) by \( (x + 3) \).
  - To make \( x + 3 \) into the LCM, you need to multiply \( x + 3 \) by \( x \).

### (b) \( x^2 + 6x - 16 \) and \( x^2 + 9x + 14 \)

First, factor both quadratic expressions.

- \( x^2 + 6x - 16 \) factors as \( (x - 2)(x + 8) \).
- \( x^2 + 9x + 14 \) factors as \( (x + 2)(x + 7) \).

- **LCM of \( x^2 + 6x - 16 \) and \( x^2 + 9x + 14 \):**
  To find the LCM, we take the product of all distinct linear factors, raised to their highest powers:

  \[
  \text{LCM}(x^2 + 6x - 16, x^2 + 9x + 14) = (x - 2)(x + 8)(x + 2)(x + 7)
  \]

- **Multiplying factors:**
  - To make \( x^2 + 6x - 16 \) into the LCM, you need to multiply it by \( (x + 2)(x + 7) \).
  - To make \( x^2 + 9x + 14 \) into the LCM, you need to multiply it by \( (x - 2)(x + 8) \).

### (c) \( x^3 - 36x \) and \( x^2 + 2x - 24 \)

First, factor both expressions.

- \( x^3 - 36x = x(x - 6)(x + 6) \).
- \( x^2 + 2x - 24 \) factors as \( (x - 6)(x + 4) \).

- **LCM of \( x^3 - 36x \) and \( x^2 + 2x - 24 \):**
  To find the LCM, we take the product of all distinct linear factors, raised to their highest powers:

  \[
  \text{LCM}(x^3 - 36x, x^2 + 2x - 24) = x(x - 6)(x + 6)(x + 4)
  \]

- **Multiplying factors:**
  - To make \( x^3 - 36x \) into the LCM, you need to multiply it by \( (x + 4) \).
  - To make \( x^2 + 2x - 24 \) into the LCM, you need to multiply it by \( x(x + 6) \).

### (d) \( x^2 + 2tx - 3t^2 \) and \( x^2 - t^2 \)

First, factor both expressions.

- \( x^2 + 2tx - 3t^2 \) factors as \( (x + 3t)(x - t) \).
- \( x^2 - t^2 \) is a difference of squares and factors as \( (x + t)(x - t) \).

- **LCM of \( x^2 + 2tx - 3t^2 \) and \( x^2 - t^2 \):**
  To find the LCM, we take the product of all distinct linear factors, raised to their highest powers:

  \[
  \text{LCM}(x^2 + 2tx - 3t^2, x^2 - t^2) = (x + 3t)(x + t)(x - t)
  \]

- **Multiplying factors:**
  - To make \( x^2 + 2tx - 3t^2 \) into the LCM, you need to multiply it by \( (x + t) \).
  - To make \( x^2 - t^2 \) into the LCM, you need to multiply it by \( (x + 3t) \).

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Here are 5 questions related to the problem:
1. What is the difference between the least common multiple and greatest common divisor?
2. How do you factor cubic polynomials like \( x^3 - 36x \)?
3. Why is factoring necessary when determining the least common multiple of polynomials?
4. Can a least common multiple of two polynomials be simpler than their product?
5. How can the LCM concept be extended to more than two polynomials?

**Tip:** Always check the factorization of each expression carefully to ensure accuracy when calculating the LCM!

Find the least common multiple of the expressions below. Once found, determine the quantities with which you need to multiply each expression in order to obtain the least common multiple.
(a) x and x + 3.
(b) x^2 + 6x − 16 and x^2 + 9x + 14.
(c) x^3 − 36x and x^2 + 2x − 24.
(d) x^2 + 2tx − 3t^2 and x^2 − t^2.

Learn how to find the least common multiple (LCM) of polynomial expressions such as x, x + 3, x^2 + 6x − 16, and more. This detailed solution covers factoring techniques, polynomial LCM calculations, and relevant algebraic concepts. Suitable for high school students in Grades 10-12.

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