The polynomiaal `(a x^2+b x+c)(a x^2-dx-c), a c!=0` has

Options

A. Four real zeros
B. At least two - real zeros
C. At most two-real zeros
D. No real zeros

Answer

To solve the problem, we need to analyze the polynomial given by the product of two quadratic expressions:

\[ (ax^2 + bx + c)(ax^2 - dx - c) \]

Given that \( ac \neq 0 \), we can conclude that both \( a \) and \( c \) are non-zero.

### Step 1: Identify the form of the polynomial
The polynomial can be expanded as follows:

\[ P(x) = (ax^2 + bx + c)(ax^2 - dx - c) \]

### Step 2: Expand the polynomial
Using the distributive property (FOIL method), we can expand the polynomial:

\[ P(x) = ax^2(ax^2) + ax^2(-dx) + ax^2(-c) + bx(ax^2) + bx(-dx) + bx(-c) + c(ax^2) + c(-dx) + c(-c) \]

This simplifies to:

\[ P(x) = a^2x^4 + (b - d)ax^3 + (c - bd - ac)x^2 - (bc + dc)x - c^2 \]

### Step 3: Analyze the coefficients
The polynomial \( P(x) \) is a quartic polynomial (degree 4). The coefficients of the polynomial are:

- Coefficient of \( x^4 \): \( a^2 \)
- Coefficient of \( x^3 \): \( (b - d)a \)
- Coefficient of \( x^2 \): \( (c - bd - ac) \)
- Coefficient of \( x^1 \): \( -(bc + dc) \)
- Constant term: \( -c^2 \)

### Step 4: Determine the nature of the roots
To determine the nature of the roots of the polynomial, we can use the discriminant method for quadratic equations.

1. For the first quadratic \( ax^2 + bx + c \):
- Discriminant \( D_1 = b^2 - 4ac \)

2. For the second quadratic \( ax^2 - dx - c \):
- Discriminant \( D_2 = (-d)^2 - 4a(-c) = d^2 + 4ac \)

Since \( ac \neq 0 \), we analyze the signs of \( D_1 \) and \( D_2 \):

- If \( ac > 0 \), then \( D_2 > 0 \) implies that the second quadratic has two real roots.
- The first quadratic may have either two real roots or none depending on the value of \( D_1 \).

### Step 5: Conclusion about the roots of the polynomial
Since at least one of the quadratics has real roots, we conclude that the polynomial \( P(x) \) has at least two real roots.

### Final Answer
The polynomial \( (ax^2 + bx + c)(ax^2 - dx - c) \) has at least two real roots.

---