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Answer :
We start with the polynomial
[tex]$$
5x^3 - 45x^2 + 70x.
$$[/tex]
Step 1. Factor out the common factor
Notice that each term in the polynomial has a common factor of [tex]$5x$[/tex]. Factor this out:
[tex]$$
5x^3 - 45x^2 + 70x = 5x (x^2 - 9x + 14).
$$[/tex]
Step 2. Factor the quadratic
Now, we need to factor the quadratic expression inside the parentheses:
[tex]$$
x^2 - 9x + 14.
$$[/tex]
To factor this, look for two numbers that multiply to [tex]$14$[/tex] (the constant term) and add up to [tex]$-9$[/tex] (the coefficient of [tex]$x$[/tex]). These numbers are [tex]$-2$[/tex] and [tex]$-7$[/tex], since
[tex]$$
(-2) \times (-7) = 14 \quad \text{and} \quad (-2) + (-7) = -9.
$$[/tex]
Therefore, we can factor the quadratic as:
[tex]$$
x^2 - 9x + 14 = (x - 2)(x - 7).
$$[/tex]
Step 3. Write the fully factored form
Substitute the factored quadratic back into the expression:
[tex]$$
5x^3 - 45x^2 + 70x = 5x (x - 2)(x - 7).
$$[/tex]
Thus, the complete factorization of the polynomial is
[tex]$$
5x (x - 2)(x - 7).
$$[/tex]
[tex]$$
5x^3 - 45x^2 + 70x.
$$[/tex]
Step 1. Factor out the common factor
Notice that each term in the polynomial has a common factor of [tex]$5x$[/tex]. Factor this out:
[tex]$$
5x^3 - 45x^2 + 70x = 5x (x^2 - 9x + 14).
$$[/tex]
Step 2. Factor the quadratic
Now, we need to factor the quadratic expression inside the parentheses:
[tex]$$
x^2 - 9x + 14.
$$[/tex]
To factor this, look for two numbers that multiply to [tex]$14$[/tex] (the constant term) and add up to [tex]$-9$[/tex] (the coefficient of [tex]$x$[/tex]). These numbers are [tex]$-2$[/tex] and [tex]$-7$[/tex], since
[tex]$$
(-2) \times (-7) = 14 \quad \text{and} \quad (-2) + (-7) = -9.
$$[/tex]
Therefore, we can factor the quadratic as:
[tex]$$
x^2 - 9x + 14 = (x - 2)(x - 7).
$$[/tex]
Step 3. Write the fully factored form
Substitute the factored quadratic back into the expression:
[tex]$$
5x^3 - 45x^2 + 70x = 5x (x - 2)(x - 7).
$$[/tex]
Thus, the complete factorization of the polynomial is
[tex]$$
5x (x - 2)(x - 7).
$$[/tex]
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