Answer :
To factor the polynomial [tex]\(-9x^5 - 33x^4 - 60x^3\)[/tex] completely, we'll go through a step-by-step process:
1. Identify a Common Factor: First, we look for any common factors in all terms of the polynomial. The terms are [tex]\(-9x^5\)[/tex], [tex]\(-33x^4\)[/tex], and [tex]\(-60x^3\)[/tex].
2. Factor Out the Greatest Common Factor (GCF): The greatest common factor of the coefficients [tex]\(-9\)[/tex], [tex]\(-33\)[/tex], and [tex]\(-60\)[/tex] is [tex]\(-3\)[/tex]. Additionally, each term has at least [tex]\(x^3\)[/tex] as a factor. Therefore, we can factor out [tex]\(-3x^3\)[/tex].
[tex]\[
-9x^5 - 33x^4 - 60x^3 = -3x^3(3x^2 + 11x + 20)
\][/tex]
3. Factor the Remaining Polynomial: Next, we need to see if the quadratic polynomial [tex]\(3x^2 + 11x + 20\)[/tex] can be factored further.
4. Check for Factorability: To factor [tex]\(3x^2 + 11x + 20\)[/tex], we need to find two numbers that multiply to [tex]\(3 \times 20 = 60\)[/tex] and add up to [tex]\(11\)[/tex]. The numbers 5 and 6 fit this criterion since [tex]\(5 \times 12 = 60\)[/tex] and [tex]\(5 + 12 = 17\)[/tex].
5. Factoring the Quadratic: Since no simple factor pair was found in step 4 to satisfy the condition, [tex]\(3x^2 + 11x + 20\)[/tex] does not factor over the integers, indicating this is already in its simplest factored form along with the GCF.
So, the polynomial is completely factored as:
[tex]\[
-3x^3(3x^2 + 11x + 20)
\][/tex]
1. Identify a Common Factor: First, we look for any common factors in all terms of the polynomial. The terms are [tex]\(-9x^5\)[/tex], [tex]\(-33x^4\)[/tex], and [tex]\(-60x^3\)[/tex].
2. Factor Out the Greatest Common Factor (GCF): The greatest common factor of the coefficients [tex]\(-9\)[/tex], [tex]\(-33\)[/tex], and [tex]\(-60\)[/tex] is [tex]\(-3\)[/tex]. Additionally, each term has at least [tex]\(x^3\)[/tex] as a factor. Therefore, we can factor out [tex]\(-3x^3\)[/tex].
[tex]\[
-9x^5 - 33x^4 - 60x^3 = -3x^3(3x^2 + 11x + 20)
\][/tex]
3. Factor the Remaining Polynomial: Next, we need to see if the quadratic polynomial [tex]\(3x^2 + 11x + 20\)[/tex] can be factored further.
4. Check for Factorability: To factor [tex]\(3x^2 + 11x + 20\)[/tex], we need to find two numbers that multiply to [tex]\(3 \times 20 = 60\)[/tex] and add up to [tex]\(11\)[/tex]. The numbers 5 and 6 fit this criterion since [tex]\(5 \times 12 = 60\)[/tex] and [tex]\(5 + 12 = 17\)[/tex].
5. Factoring the Quadratic: Since no simple factor pair was found in step 4 to satisfy the condition, [tex]\(3x^2 + 11x + 20\)[/tex] does not factor over the integers, indicating this is already in its simplest factored form along with the GCF.
So, the polynomial is completely factored as:
[tex]\[
-3x^3(3x^2 + 11x + 20)
\][/tex]
