Assertion (A) : If 1,2,3 are the roots of `ax^3 + bx ^2 +cx +d=0` then the roots of ` ax^3 +2bx^2 + 4 cx + 8d =0` are ` 2,4,6`
Reason (R ) : the equation whose roots are k times the roots of the equation `f(x) =0` is ` f((x )/(k ))=0`

Options

A. both A and R are true R is the correct explanation of A
B. both A and R are true and R is not correct explanation of A
C. A is true and R is false
D. A is false and R is true

Answer

To solve the given assertion and reason, we will analyze the statements step by step.

### Step 1: Understand the Assertion
The assertion states that if \(1, 2, 3\) are the roots of the polynomial equation \(ax^3 + bx^2 + cx + d = 0\), then the roots of the polynomial \(ax^3 + 2bx^2 + 4cx + 8d = 0\) are \(2, 4, 6\).

### Step 2: Identify the Roots of the Original Polynomial
The roots of the polynomial \(ax^3 + bx^2 + cx + d = 0\) are given as \(1, 2, 3\).

### Step 3: Determine the New Roots
If we multiply each root of the original polynomial by \(2\), we get:
- \(2 \times 1 = 2\)
- \(2 \times 2 = 4\)
- \(2 \times 3 = 6\)

Thus, the new roots are indeed \(2, 4, 6\).

### Step 4: Analyze the Reason
The reason states that the equation whose roots are \(k\) times the roots of the equation \(f(x) = 0\) is given by \(f\left(\frac{x}{k}\right) = 0\).

### Step 5: Apply the Reason to the Given Polynomial
In our case, \(k = 2\). Therefore, we need to evaluate \(f\left(\frac{x}{2}\right)\):
1. Start with \(f(x) = ax^3 + bx^2 + cx + d\).
2. Substitute \(x\) with \(\frac{x}{2}\):
\[ f\left(\frac{x}{2}\right) = a\left(\frac{x}{2}\right)^3 + b\left(\frac{x}{2}\right)^2 + c\left(\frac{x}{2}\right) + d \]
\[ = a\frac{x^3}{8} + b\frac{x^2}{4} + c\frac{x}{2} + d \]

### Step 6: Clear the Denominator
To eliminate the fractions, multiply the entire equation by \(8\):
\[ 8f\left(\frac{x}{2}\right) = ax^3 + 2bx^2 + 4cx + 8d = 0 \]

### Step 7: Conclusion
The new polynomial \(ax^3 + 2bx^2 + 4cx + 8d = 0\) indeed has roots \(2, 4, 6\), confirming the assertion is true. The reason provided is also valid as it correctly explains the relationship between the roots of the two equations.

### Final Answer
Both the assertion (A) and the reason (R) are true, and R is the correct explanation of A.