Question

1. $x^4 - 12x^2 + 36$
2. w
3. $k^3 + 7k^2 - 44k$

$k(k^2 + 7k - 44)$
$k(k + 11)(k - 4)$
17.

1. $-x^3 + 4x^2 + 21x$

$-x(x^2 - 4x - 21)$
$-x(x - 7)(x + 3)$
19.

1. $9y^6 + 6y^4 + y^2$

Explanation:

Let's solve each problem one by one:

Problem 14: \( x^4 - 12x^2 + 36 \)

Step 1: Recognize as a quadratic in terms of \( x^2 \)

Let \( u = x^2 \), then the expression becomes \( u^2 - 12u + 36 \).

Step 2: Factor the quadratic

We need two numbers that multiply to \( 36 \) and add to \( -12 \). Those numbers are \( -6 \) and \( -6 \). So, \( u^2 - 12u + 36 = (u - 6)^2 \).

Step 3: Substitute back \( u = x^2 \)

\( (x^2 - 6)^2 \)

Step 1: Factor out the greatest common factor (GCF)

The GCF of \( k^3 \), \( 7k^2 \), and \( -44k \) is \( k \). So, \( k(k^2 + 7k - 44) \).

Step 2: Factor the quadratic \( k^2 + 7k - 44 \)

We need two numbers that multiply to \( -44 \) and add to \( 7 \). Those numbers are \( 11 \) and \( -4 \). So, \( k^2 + 7k - 44 = (k + 11)(k - 4) \).

Step 3: Combine the factors

\( k(k + 11)(k - 4) \)

Step 1: Factor out the GCF (including the negative sign)

Factor out \( -x \) from \( -x^3 \), \( 4x^2 \), and \( 21x \). So, \( -x(x^2 - 4x - 21) \).

Step 2: Factor the quadratic \( x^2 - 4x - 21 \)

We need two numbers that multiply to \( -21 \) and add to \( -4 \). Those numbers are \( -7 \) and \( 3 \). So, \( x^2 - 4x - 21 = (x - 7)(x + 3) \).

Step 3: Combine the factors

\( -x(x - 7)(x + 3) \)

Answer:

\( (x^2 - 6)^2 \)

Problem 16: \( k^3 + 7k^2 - 44k \)