If `(x-a)` is a factor of polynomial `p(x)` then `a` is a zero of `a xx p(x)`.True Or False

Answer

To determine whether the statement "If `(x-a)` is a factor of polynomial `p(x)`, then `a` is a zero of `a * p(x)`" is true or false, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Factor**: We are given that `(x-a)` is a factor of the polynomial `p(x)`. According to the Factor Theorem, if `(x-a)` is a factor of `p(x)`, then substituting `x = a` into `p(x)` should yield zero. Therefore, we have:
\[ p(a) = 0 \]

2. **Defining the New Polynomial**: We need to analyze the polynomial `h(x) = a * p(x)`. Here, `a` is a constant and `p(x)` is a polynomial.

3. **Substituting `x = a` into `h(x)`**: We will check if `a` is a zero of `h(x)` by substituting `x = a`:
\[ h(a) = a * p(a) \]

4. **Using the Result from Step 1**: From Step 1, we know that `p(a) = 0`. Therefore, substituting this into our equation for `h(a)` gives:
\[ h(a) = a * p(a) = a * 0 = 0 \]

5. **Conclusion**: Since `h(a) = 0`, we conclude that `a` is indeed a zero of the polynomial `h(x) = a * p(x)`.

Thus, the statement is **True**.