Question

factor ( x^4 - 10x^2 + 9 ) completely. all factors in your answer should have integer coefficients.

Explanation:

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

The given polynomial is \( x^4 - 10x^2 + 9 \). Let \( y = x^2 \), then the polynomial becomes \( y^2 - 10y + 9 \).

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

We need to find two numbers that multiply to \( 9 \) and add up to \( -10 \). The numbers are \( -1 \) and \( -9 \). So, \( y^2 - 10y + 9=(y - 1)(y - 9) \).

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

Substituting \( y = x^2 \) back, we get \( (x^2 - 1)(x^2 - 9) \).

Step4: Factor the difference of squares.

We know 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 \( x^2 - 9 \), \( a = x \), \( b = 3 \), so \( x^2 - 9=(x - 3)(x + 3) \).

Step5: Combine all factors.

Putting it all together, \( x^4 - 10x^2 + 9=(x - 1)(x + 1)(x - 3)(x + 3) \).

Answer:

\((x - 1)(x + 1)(x - 3)(x + 3)\)