(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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Let me know if you want further details or have any questions!

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!