If the lcm of the polynomials x9+x6+x3+1 and x6-1 is x12-1, then their hcf is

To find the highest common factor (HCF) of the polynomials \(x^9 + x^6 + x^3 + 1\) and \(x^6 - 1\), given that their least common multiple (LCM) is \(x^{12} - 1\), we can use the relationship between LCM and HCF. This relationship states that the product of the LCM and HCF of two polynomials is equal to the product of the polynomials themselves. In mathematical terms, we can express this as:

Finding the HCF Using the Relationship

Let’s denote the given polynomials as:

• Let \(A = x^9 + x^6 + x^3 + 1\)
• Let \(B = x^6 - 1\)

According to the relationship:

LCM(A, B) × HCF(A, B) = A × B

From the problem, we know:

• LCM(A, B) = \(x^{12} - 1\)

Now, we can express the relationship with the known values:

(x^{12} - 1) × HCF(A, B) = A × B

Calculating the Product of A and B

Next, we need to calculate \(A \times B\):

A × B = (x^9 + x^6 + x^3 + 1) × (x^6 - 1)

To simplify this, we can expand the expression:

• First, distribute \(A\) over \(B\):
• We get \(x^9(x^6 - 1) + x^6(x^6 - 1) + x^3(x^6 - 1) + 1(x^6 - 1)\)

Calculating each term gives:

• \(x^{15} - x^9\)
• \(x^{12} - x^6\)
• \(x^9 - x^3\)
• \(x^6 - 1\)

Now, combining all these terms, we have:

A × B = x^{15} + x^{12} + x^9 + x^6 - (x^9 + x^6 + 1) = x^{15} + x^{12} - 1

Setting Up the HCF Equation

Substituting back into our earlier equation, we have:

(x^{12} - 1) × HCF(A, B) = (x^{15} + x^{12} - 1)

To find \(HCF(A, B)\), we can isolate it:

HCF(A, B) = (x^{15} + x^{12} - 1) / (x^{12} - 1)

Dividing the Polynomials

Now, we need to perform the polynomial division of \(x^{15} + x^{12} - 1\) by \(x^{12} - 1\). When we do this division, the leading term \(x^{15}\) divided by \(x^{12}\) gives \(x^3\). Multiplying back and subtracting will give us the remainder.

After performing the division, we find:

HCF(A, B) = x^3 + 1

This means that the highest common factor of the polynomials \(x^9 + x^6 + x^3 + 1\) and \(x^6 - 1\) is \(x^3 + 1\).

Final Result

To summarize, the HCF of the given polynomials is:

HCF = x^3 + 1