Question

factor ( x^4 - 7x^2 - 18 ) completely. all factors in your answer should have integer coefficients.

Explanation:

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

The given expression is \( x^4 - 7x^2 - 18 \). Let \( y = x^2 \), then the expression becomes \( y^2 - 7y - 18 \).

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

We need to find two numbers that multiply to \( - 18 \) and add up to \( - 7 \). The numbers are \( - 9 \) and \( 2 \) since \( - 9\times2=-18 \) and \( - 9 + 2=-7 \). So, \( y^2 - 7y - 18=(y - 9)(y + 2) \).

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

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

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

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

Step5: Combine all factors.

Putting it all together, \( x^4 - 7x^2 - 18=(x - 3)(x + 3)(x^2 + 2) \).

Answer:

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