Question

factor ( x^4 - 5x^2 + 4 ) completely. all factors in your answer should have integer coefficients.

Explanation:

Step1: Let \( y = x^2 \), rewrite the polynomial.

The given polynomial is \( x^4 - 5x^2 + 4 \). Substitute \( y = x^2 \), then the polynomial becomes \( y^2 - 5y + 4 \).

Step2: Factor the quadratic in \( y \).

We need to find two numbers that multiply to \( 4 \) and add up to \( -5 \). The numbers are \( -1 \) and \( -4 \). So, \( y^2 - 5y + 4=(y - 1)(y - 4) \).

Step3: Substitute back \( y = x^2 \).

Substituting \( y = x^2 \) into \( (y - 1)(y - 4) \), we get \( (x^2 - 1)(x^2 - 4) \).

Step4: Factor the difference of squares.

Recall that \( a^2 - b^2=(a - b)(a + b) \). For \( x^2 - 1 \), \( a = x \), \( b = 1 \), so \( x^2 - 1=(x - 1)(x + 1) \). For \( x^2 - 4 \), \( a = x \), \( b = 2 \), so \( x^2 - 4=(x - 2)(x + 2) \).

Step5: Combine all factors.

Putting it all together, \( x^4 - 5x^2 + 4=(x - 1)(x + 1)(x - 2)(x + 2) \).

Answer:

\((x - 1)(x + 1)(x - 2)(x + 2)\)