Answer :
To factor the polynomial [tex]\(x^5 - 60x^3 - 7x^4\)[/tex] completely, follow these steps:
1. Look for a common factor:
- Notice that each term of the polynomial contains at least [tex]\(x^3\)[/tex]. So, [tex]\(x^3\)[/tex] is a common factor.
- Factor [tex]\(x^3\)[/tex] out of the polynomial:
[tex]\[
x^5 - 60x^3 - 7x^4 = x^3(x^2 - 7x - 60)
\][/tex]
2. Factor the quadratic expression:
- Now, focus on factoring the quadratic part: [tex]\(x^2 - 7x - 60\)[/tex].
- To factor this, look for two numbers that multiply to [tex]\(-60\)[/tex] (the constant term) and add to [tex]\(-7\)[/tex] (the coefficient of the linear term).
- These two numbers are [tex]\(-12\)[/tex] and [tex]\(5\)[/tex] because [tex]\(-12 \times 5 = -60\)[/tex] and [tex]\(-12 + 5 = -7\)[/tex].
3. Write the factored form:
- The quadratic can be factored as [tex]\((x - 12)(x + 5)\)[/tex].
4. Combine everything:
- Finally, substitute back to write the complete factorization of the original polynomial:
[tex]\[
x^5 - 60x^3 - 7x^4 = x^3(x - 12)(x + 5)
\][/tex]
So, the completely factored form of the polynomial is [tex]\(x^3(x - 12)(x + 5)\)[/tex].
1. Look for a common factor:
- Notice that each term of the polynomial contains at least [tex]\(x^3\)[/tex]. So, [tex]\(x^3\)[/tex] is a common factor.
- Factor [tex]\(x^3\)[/tex] out of the polynomial:
[tex]\[
x^5 - 60x^3 - 7x^4 = x^3(x^2 - 7x - 60)
\][/tex]
2. Factor the quadratic expression:
- Now, focus on factoring the quadratic part: [tex]\(x^2 - 7x - 60\)[/tex].
- To factor this, look for two numbers that multiply to [tex]\(-60\)[/tex] (the constant term) and add to [tex]\(-7\)[/tex] (the coefficient of the linear term).
- These two numbers are [tex]\(-12\)[/tex] and [tex]\(5\)[/tex] because [tex]\(-12 \times 5 = -60\)[/tex] and [tex]\(-12 + 5 = -7\)[/tex].
3. Write the factored form:
- The quadratic can be factored as [tex]\((x - 12)(x + 5)\)[/tex].
4. Combine everything:
- Finally, substitute back to write the complete factorization of the original polynomial:
[tex]\[
x^5 - 60x^3 - 7x^4 = x^3(x - 12)(x + 5)
\][/tex]
So, the completely factored form of the polynomial is [tex]\(x^3(x - 12)(x + 5)\)[/tex].
