Question

factor ( x^4 - 13x^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 \( x^4 - 13x^2 + 36 \). Substitute \( y = x^2 \), then the polynomial becomes \( y^2 - 13y + 36 \).

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

We need to find two numbers that multiply to \( 36 \) and add up to \( -13 \). The numbers are \( -4 \) and \( -9 \) because \( (-4)\times(-9)=36 \) and \( (-4)+(-9)=-13 \). So, \( y^2 - 13y + 36=(y - 4)(y - 9) \).

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

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

Step4: Factor the difference of squares.

Recall that \( a^2 - b^2=(a - b)(a + b) \). For \( x^2 - 4 \), \( a = x \), \( b = 2 \), so \( x^2 - 4=(x - 2)(x + 2) \). For \( x^2 - 9 \), \( a = x \), \( b = 3 \), so \( x^2 - 9=(x - 3)(x + 3) \).

Step5: Combine all factors.

Putting it all together, \( x^4 - 13x^2 + 36=(x - 2)(x + 2)(x - 3)(x + 3) \).

Answer:

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