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Answer :
We start with the polynomial
[tex]$$10x^3 - 25x^2 - 35x.$$[/tex]
Step 1. Factor out the greatest common factor (GCF).
Each term of the polynomial has a factor of [tex]$5x$[/tex], so we factor [tex]$5x$[/tex] out:
[tex]$$10x^3 - 25x^2 - 35x = 5x(2x^2 - 5x - 7).$$[/tex]
Step 2. Factor the quadratic expression.
Next, we focus on factoring the quadratic
[tex]$$2x^2 - 5x - 7.$$[/tex]
To factor this quadratic, we look for two numbers that multiply to [tex]$2 \times (-7) = -14$[/tex] and add to [tex]$-5$[/tex]. Observing the factors, we find that the quadratic factors as:
[tex]$$(x + 1)(2x - 7).$$[/tex]
You can verify this by expanding the product:
[tex]$$
(x + 1)(2x - 7) = 2x^2 - 7x + 2x - 7 = 2x^2 - 5x - 7.
$$[/tex]
Step 3. Write the complete factorization.
Substitute the factored form of the quadratic back into the expression we obtained after factoring out the GCF:
[tex]$$10x^3 - 25x^2 - 35x = 5x(2x^2 - 5x - 7) = 5x(x + 1)(2x - 7).$$[/tex]
Thus, the fully factored form of the polynomial is
[tex]$$5x(x + 1)(2x - 7).$$[/tex]
[tex]$$10x^3 - 25x^2 - 35x.$$[/tex]
Step 1. Factor out the greatest common factor (GCF).
Each term of the polynomial has a factor of [tex]$5x$[/tex], so we factor [tex]$5x$[/tex] out:
[tex]$$10x^3 - 25x^2 - 35x = 5x(2x^2 - 5x - 7).$$[/tex]
Step 2. Factor the quadratic expression.
Next, we focus on factoring the quadratic
[tex]$$2x^2 - 5x - 7.$$[/tex]
To factor this quadratic, we look for two numbers that multiply to [tex]$2 \times (-7) = -14$[/tex] and add to [tex]$-5$[/tex]. Observing the factors, we find that the quadratic factors as:
[tex]$$(x + 1)(2x - 7).$$[/tex]
You can verify this by expanding the product:
[tex]$$
(x + 1)(2x - 7) = 2x^2 - 7x + 2x - 7 = 2x^2 - 5x - 7.
$$[/tex]
Step 3. Write the complete factorization.
Substitute the factored form of the quadratic back into the expression we obtained after factoring out the GCF:
[tex]$$10x^3 - 25x^2 - 35x = 5x(2x^2 - 5x - 7) = 5x(x + 1)(2x - 7).$$[/tex]
Thus, the fully factored form of the polynomial is
[tex]$$5x(x + 1)(2x - 7).$$[/tex]
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