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Factor the following polynomial completely.

[tex]\[ 5x^4 + 20x^3 - 105x^2 \][/tex]

A. [tex]\[5x^2(x+3)(x-7)\][/tex]

B. [tex]\[5x^2(x-3)(x+7)\][/tex]

C. [tex]\[5x^3(x-3)(x+7)\][/tex]

D. [tex]\[5x^3(x+3)(x-7)\][/tex]


Sagot :

Sure, let's factor the polynomial [tex]\( 5x^4 + 20x^3 - 105x^2 \)[/tex] completely, step by step.

1. Identify common factors:
Look for any common factor in all terms of the polynomial. The terms are [tex]\( 5x^4 \)[/tex], [tex]\( 20x^3 \)[/tex], and [tex]\( -105x^2 \)[/tex].

All terms have a common factor of [tex]\( 5x^2 \)[/tex].
So, factor out [tex]\( 5x^2 \)[/tex]:
[tex]\[ 5x^2 (x^2 + 4x - 21) \][/tex]

2. Factor the quadratic expression:
Now, we focus on factoring the quadratic expression [tex]\( x^2 + 4x - 21 \)[/tex].

To factor the quadratic expression, we need to find two numbers that multiply to the constant term (-21) and add to the coefficient of the middle term (4).

- The constant term is -21.
- The middle term's coefficient is 4.

The two numbers that fit these criteria are 7 and -3:
[tex]\[ 7 \times (-3) = -21 \][/tex]
[tex]\[ 7 + (-3) = 4 \][/tex]

So, the quadratic expression [tex]\( x^2 + 4x - 21 \)[/tex] can be factored as:
[tex]\[ (x + 7)(x - 3) \][/tex]

3. Write the complete factorization:
Substituting this back into our original factorization:
[tex]\[ 5x^2 (x + 7)(x - 3) \][/tex]

Thus, the completely factored form of the polynomial [tex]\( 5x^4 + 20x^3 - 105x^2 \)[/tex] is:
[tex]\[ 5x^2(x - 3)(x + 7) \][/tex]

The correct answer is:
[tex]\[ \boxed{\text{B. } 5x^2(x-3)(x+7)} \][/tex]