Question

The polynomial \[ p(x)=x^4-5x^2+4 \] has:

• A. Four distinct real roots
• B. Two distinct real roots
• C. No real roots
• D. One repeated real root

Hint:

For even degree polynomials involving only even powers of \(x\), substitute: \[ y=x^2 \] to reduce the equation into a simpler quadratic form.

Answer 1

The Correct Option is A

Concept: To determine the number of real roots of a polynomial, factorization is usually the most efficient method. If a quartic polynomial can be reduced into quadratic factors, its roots can be obtained directly.

Step 1: Rewrite the polynomial in quadratic form. Given: \[ p(x)=x^4-5x^2+4 \] Let: \[ y=x^2 \] Then the equation becomes: \[ y^2-5y+4 \] Factorize: \[ y^2-5y+4=(y-1)(y-4) \] Substituting back \(y=x^2\): \[ (x^2-1)(x^2-4) \] Further factorizing: \[ (x-1)(x+1)(x-2)(x+2) \]

Step 2: Find all roots. The roots are: \[ x=1,\quad x=-1,\quad x=2,\quad x=-2 \] All roots are:
• Real
• Distinct Hence the polynomial has four distinct real roots. Therefore, \[ \boxed{ \text{Four distinct real roots} } \] Hence the correct answer is: \[ \boxed{(A)} \]