Answer :
We start with the expression
[tex]$$-4x^9 - 6x^8 + 22x^6.$$[/tex]
Step 1. Identify the numerical coefficients and the exponents of [tex]\( x \)[/tex]:
- The coefficients are [tex]\(-4\)[/tex], [tex]\(-6\)[/tex], and [tex]\(22\)[/tex].
- The exponents are [tex]\(9\)[/tex], [tex]\(8\)[/tex], and [tex]\(6\)[/tex].
Step 2. Find the greatest common divisor (GCD) of the coefficients. The GCD of the absolute values [tex]\(4\)[/tex], [tex]\(6\)[/tex], and [tex]\(22\)[/tex] is [tex]\(2\)[/tex].
Step 3. Determine the smallest exponent. Among [tex]\(9\)[/tex], [tex]\(8\)[/tex], and [tex]\(6\)[/tex], the smallest exponent is [tex]\(6\)[/tex].
Step 4. Form the greatest common factor (GCF) including a negative sign. Since we want the GCF to include a negative, we factor out [tex]\(-2x^6\)[/tex].
Step 5. Divide each term by [tex]\(-2x^6\)[/tex]:
- For the first term:
[tex]$$
\frac{-4x^9}{-2x^6} = 2x^{9-6} = 2x^3.
$$[/tex]
- For the second term:
[tex]$$
\frac{-6x^8}{-2x^6} = 3x^{8-6} = 3x^2.
$$[/tex]
- For the third term:
[tex]$$
\frac{22x^6}{-2x^6} = -11.
$$[/tex]
Step 6. Write the factorization by placing the GCF outside of parentheses:
[tex]$$
-4x^9 - 6x^8 + 22x^6 = -2x^6(2x^3 + 3x^2 - 11).
$$[/tex]
Thus, the fully factored form of the expression is
[tex]$$-2x^6(2x^3 + 3x^2 - 11).$$[/tex]
[tex]$$-4x^9 - 6x^8 + 22x^6.$$[/tex]
Step 1. Identify the numerical coefficients and the exponents of [tex]\( x \)[/tex]:
- The coefficients are [tex]\(-4\)[/tex], [tex]\(-6\)[/tex], and [tex]\(22\)[/tex].
- The exponents are [tex]\(9\)[/tex], [tex]\(8\)[/tex], and [tex]\(6\)[/tex].
Step 2. Find the greatest common divisor (GCD) of the coefficients. The GCD of the absolute values [tex]\(4\)[/tex], [tex]\(6\)[/tex], and [tex]\(22\)[/tex] is [tex]\(2\)[/tex].
Step 3. Determine the smallest exponent. Among [tex]\(9\)[/tex], [tex]\(8\)[/tex], and [tex]\(6\)[/tex], the smallest exponent is [tex]\(6\)[/tex].
Step 4. Form the greatest common factor (GCF) including a negative sign. Since we want the GCF to include a negative, we factor out [tex]\(-2x^6\)[/tex].
Step 5. Divide each term by [tex]\(-2x^6\)[/tex]:
- For the first term:
[tex]$$
\frac{-4x^9}{-2x^6} = 2x^{9-6} = 2x^3.
$$[/tex]
- For the second term:
[tex]$$
\frac{-6x^8}{-2x^6} = 3x^{8-6} = 3x^2.
$$[/tex]
- For the third term:
[tex]$$
\frac{22x^6}{-2x^6} = -11.
$$[/tex]
Step 6. Write the factorization by placing the GCF outside of parentheses:
[tex]$$
-4x^9 - 6x^8 + 22x^6 = -2x^6(2x^3 + 3x^2 - 11).
$$[/tex]
Thus, the fully factored form of the expression is
[tex]$$-2x^6(2x^3 + 3x^2 - 11).$$[/tex]
