Question

factor ( x^4 - 5x^2 - 36 ) completely. all factors in your answer should have integer coefficients.

Explanation:

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

The given polynomial is \( x^4 - 5x^2 - 36 \). Let \( y = x^2 \), then the polynomial becomes \( y^2 - 5y - 36 \).

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

We need to find two numbers that multiply to \( - 36 \) and add up to \( - 5 \). The numbers are \( - 9 \) and \( 4 \) because \( (-9)\times4=-36 \) and \( -9 + 4=-5 \). So, \( y^2 - 5y - 36=(y - 9)(y + 4) \).

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

Substituting \( y = x^2 \) into \( (y - 9)(y + 4) \), we get \( (x^2 - 9)(x^2 + 4) \).

Step4: Factor \( x^2 - 9 \) using difference of squares.

The formula for difference of squares is \( a^2 - b^2=(a - b)(a + b) \). Here, \( a = x \) and \( b = 3 \), so \( x^2 - 9=(x - 3)(x + 3) \).

Step5: Write the completely factored form.

Putting it all together, the completely factored form of \( x^4 - 5x^2 - 36 \) is \( (x - 3)(x + 3)(x^2 + 4) \).

Answer:

\((x - 3)(x + 3)(x^2 + 4)\)