The HCF of `8x^(4) - 16x^(3) - 40x^(2) + 48x` and `16x^(5) + 64x^(4) + 80x^(3) + 32x^(2)` is

Options

A. `4x(x+2)`
B. `8x(x+2)`
C. `2x(x-2)`
D. `8x(x-2)`

Answer

To find the HCF (Highest Common Factor) of the two polynomials \(8x^4 - 16x^3 - 40x^2 + 48x\) and \(16x^5 + 64x^4 + 80x^3 + 32x^2\), we will follow the steps of factorization.

### Step 1: Factor out the common terms from the first polynomial.

The first polynomial is:
\[ 8x^4 - 16x^3 - 40x^2 + 48x \]

We can factor out the greatest common factor (GCF) from the coefficients and the variable \(x\):
- The GCF of the coefficients \(8, -16, -40, 48\) is \(8\).
- The lowest power of \(x\) in all terms is \(x\).

So, we factor out \(8x\):
\[ 8x(x^3 - 2x^2 - 5x + 6) \]

### Step 2: Factor the cubic polynomial \(x^3 - 2x^2 - 5x + 6\).

To factor \(x^3 - 2x^2 - 5x + 6\), we can use the Rational Root Theorem or synthetic division. Testing for possible rational roots, we find that \(x = 2\) is a root.

Using synthetic division to divide \(x^3 - 2x^2 - 5x + 6\) by \(x - 2\):
\[ \begin{array}{r|rrrr} 2 & 1 & -2 & -5 & 6 \\ & & 2 & 0 & -10 \\ \hline & 1 & 0 & -5 & -4 \\ \end{array} \]
This gives us:
\[ x^3 - 2x^2 - 5x + 6 = (x - 2)(x^2 - 5) \]

Thus, we can write:
\[ 8x(x - 2)(x^2 - 5) \]

### Step 3: Factor the second polynomial.

The second polynomial is:
\[ 16x^5 + 64x^4 + 80x^3 + 32x^2 \]

Again, we factor out the GCF:
- The GCF of the coefficients \(16, 64, 80, 32\) is \(16\).
- The lowest power of \(x\) is \(x^2\).

So we factor out \(16x^2\):
\[ 16x^2(x^3 + 4x^2 + 5x + 2) \]

### Step 4: Factor the cubic polynomial \(x^3 + 4x^2 + 5x + 2\).

Using synthetic division again, we can test for possible roots. Testing \(x = -2\):
\[ \begin{array}{r|rrrr} -2 & 1 & 4 & 5 & 2 \\ & & -2 & -4 & -2 \\ \hline & 1 & 2 & 1 & 0 \\ \end{array} \]
This gives us:
\[ x^3 + 4x^2 + 5x + 2 = (x + 2)(x^2 + 2x + 1) = (x + 2)(x + 1)^2 \]

Thus, we can write:
\[ 16x^2(x + 2)(x + 1)^2 \]

### Step 5: Find the HCF.

Now we have the factorizations:
1. \(8x(x - 2)(x^2 - 5)\)
2. \(16x^2(x + 2)(x + 1)^2\)

The common factors are:
- The coefficient \(8\) from the first polynomial and \(16\) from the second gives \(8\).
- The common variable factor is \(x\).

Thus, the HCF is:
\[ 8x \]

### Final Answer:
The HCF of \(8x^4 - 16x^3 - 40x^2 + 48x\) and \(16x^5 + 64x^4 + 80x^3 + 32x^2\) is \(8x\).