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Answer :
To factor the trinomial [tex]\(x^5 - 2x^4 - 35x^3\)[/tex] completely, we can follow these steps:
1. Identify the Greatest Common Factor (GCF):
Look for the GCF in the terms of the polynomial. In this case, each term contains a factor of [tex]\(x^3\)[/tex].
- [tex]\(x^5\)[/tex] has a factor of [tex]\(x^3\)[/tex].
- [tex]\(2x^4\)[/tex] has a factor of [tex]\(x^3\)[/tex].
- [tex]\(35x^3\)[/tex] obviously has a factor of [tex]\(x^3\)[/tex].
So, the GCF is [tex]\(x^3\)[/tex].
2. Factor out the GCF from the polynomial:
Factor [tex]\(x^3\)[/tex] out of each term:
[tex]\[
x^5 - 2x^4 - 35x^3 = x^3(x^2 - 2x - 35)
\][/tex]
3. Factor the quadratic expression:
Now, we need to factor the quadratic expression inside the parentheses, [tex]\(x^2 - 2x - 35\)[/tex]. We are looking for two numbers that multiply to [tex]\(-35\)[/tex] (the constant term) and add to [tex]\(-2\)[/tex] (the coefficient of the middle term).
The two numbers that satisfy this are [tex]\(-7\)[/tex] and [tex]\(5\)[/tex], because:
- [tex]\(-7 \times 5 = -35\)[/tex]
- [tex]\(-7 + 5 = -2\)[/tex]
Therefore, the quadratic factors into:
[tex]\[
x^2 - 2x - 35 = (x - 7)(x + 5)
\][/tex]
4. Write the completely factored form:
Substitute the factored quadratic back into the expression with the GCF:
[tex]\[
x^5 - 2x^4 - 35x^3 = x^3(x - 7)(x + 5)
\][/tex]
Thus, the trinomial [tex]\(x^5 - 2x^4 - 35x^3\)[/tex] is completely factored as [tex]\(x^3(x - 7)(x + 5)\)[/tex].
So the correct choice is:
A. [tex]\(x^5 - 2x^4 - 35x^3 = x^3(x - 7)(x + 5)\)[/tex]
1. Identify the Greatest Common Factor (GCF):
Look for the GCF in the terms of the polynomial. In this case, each term contains a factor of [tex]\(x^3\)[/tex].
- [tex]\(x^5\)[/tex] has a factor of [tex]\(x^3\)[/tex].
- [tex]\(2x^4\)[/tex] has a factor of [tex]\(x^3\)[/tex].
- [tex]\(35x^3\)[/tex] obviously has a factor of [tex]\(x^3\)[/tex].
So, the GCF is [tex]\(x^3\)[/tex].
2. Factor out the GCF from the polynomial:
Factor [tex]\(x^3\)[/tex] out of each term:
[tex]\[
x^5 - 2x^4 - 35x^3 = x^3(x^2 - 2x - 35)
\][/tex]
3. Factor the quadratic expression:
Now, we need to factor the quadratic expression inside the parentheses, [tex]\(x^2 - 2x - 35\)[/tex]. We are looking for two numbers that multiply to [tex]\(-35\)[/tex] (the constant term) and add to [tex]\(-2\)[/tex] (the coefficient of the middle term).
The two numbers that satisfy this are [tex]\(-7\)[/tex] and [tex]\(5\)[/tex], because:
- [tex]\(-7 \times 5 = -35\)[/tex]
- [tex]\(-7 + 5 = -2\)[/tex]
Therefore, the quadratic factors into:
[tex]\[
x^2 - 2x - 35 = (x - 7)(x + 5)
\][/tex]
4. Write the completely factored form:
Substitute the factored quadratic back into the expression with the GCF:
[tex]\[
x^5 - 2x^4 - 35x^3 = x^3(x - 7)(x + 5)
\][/tex]
Thus, the trinomial [tex]\(x^5 - 2x^4 - 35x^3\)[/tex] is completely factored as [tex]\(x^3(x - 7)(x + 5)\)[/tex].
So the correct choice is:
A. [tex]\(x^5 - 2x^4 - 35x^3 = x^3(x - 7)(x + 5)\)[/tex]
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