If the first term and common ratio of a geometric sequence
प्रश्नहरूको समाधान गर्नुहोस् (Solve the following questions):
ल. स. पत्ता लगाउनुहोस् (Find the LCM of): and

Ask by Chen Ruiz. in Nepal

Mar 29,2025

Question

If the first term and common ratio of a geometric sequence
प्रश्नहरूको समाधान गर्नुहोस् (Solve the following questions):
ल. स. पत्ता लगाउनुहोस् (Find the LCM of): and

Ask by Chen Ruiz. in Nepal

Mar 29,2025

UpStudy AI · Accepted Answer

We start by factoring each polynomial.

**Step 1. Factor $$8x^3-1$$:**

Notice that
$$
8x^3-1=(2x)^3-1^3,
$$
which is a difference of cubes. The formula for a difference of cubes is
$$
a^3-b^3=(a-b)(a^2+ab+b^2).
$$
Here, $$a=2x$$ and $$b=1$$. Thus,
$$
8x^3-1=(2x-1)\Bigl[(2x)^2+(2x)(1)+1^2\Bigr]=(2x-1)(4x^2+2x+1).
$$

**Step 2. Factor $$16x^4+4x^2+1$$:**

First, we think of writing it as a product of quadratics. We try to express it as
$$
(4x^2+2x+1)(4x^2-2x+1).
$$
Multiply these two factors to check:
$$
\begin{aligned}
(4x^2+2x+1)(4x^2-2x+1) &= 4x^2(4x^2-2x+1) + 2x(4x^2-2x+1) +1(4x^2-2x+1) \
&= 16x^4 -8x^3+4x^2 +8x^3-4x^2+2x +4x^2-2x+1 \
&= 16x^4 + ( -8x^3+8x^3) + (4x^2-4x^2+4x^2) + (2x-2x) +1 \
&= 16x^4+4x^2+1.
\end{aligned}
$$
Thus, we have factored the second polynomial as
$$
16x^4+4x^2+1=(4x^2+2x+1)(4x^2-2x+1).
$$

**Step 3. Determine the LCM:**

The factors are:
- For $$8x^3-1=(2x-1)(4x^2+2x+1)$$
- For $$16x^4+4x^2+1=(4x^2+2x+1)(4x^2-2x+1)$$

The common factor is $$(4x^2+2x+1)$$. To form the LCM we take each factor with the highest power occurring in either factorization. Therefore, the LCM is the product of:
$$
(2x-1),\ (4x^2+2x+1),\ \text{and}\ (4x^2-2x+1).
$$
Thus, the least common multiple is
$$
\text{LCM}=(2x-1)(4x^2+2x+1)(4x^2-2x+1).
$$

Alternatively, since
$$
(4x^2+2x+1)(4x^2-2x+1)=16x^4+4x^2+1,
$$
the LCM can also be written as
$$
\text{LCM}=(2x-1)(16x^4+4x^2+1).
$$

This is the required least common multiple.
The least common multiple (LCM) of $$8x^3 - 1$$ and $$16x^4 + 4x^2 + 1$$ is $$(2x - 1)(16x^4 + 4x^2 + 1)$$.

If the first term and common ratio of a geometric sequence
प्रश्नहरूको समाधान गर्नुहोस् (Solve the following questions):
ल. स. पत्ता लगाउनुहोस् (Find the LCM of): and

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If the first term and common ratio of a geometric sequence प्रश्नहरूको समाधान गर्नुहोस् (Solve the following questions): ल. स. पत्ता लगाउनुहोस् (Find the LCM of): and

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If the first term and common ratio of a geometric sequence
प्रश्नहरूको समाधान गर्नुहोस् (Solve the following questions):
ल. स. पत्ता लगाउनुहोस् (Find the LCM of): and