Question

factor the trinomial completely.

(98x^3 + 140x^2 + 50x)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. (98x^3 + 140x^2 + 50x = \\) (factor completely.)
b. the polynomial is prime.

Explanation:

⚡ Using what you learned: factoring monomials · factoring special products

Step 1: Factor out the greatest common factor (GCF)

Find the GCF of the terms in the expression \( 98x^3 + 140x^2 + 50x \).

The GCF of the coefficients \( 98 \), \( 140 \), and \( 50 \) is \( 2 \).
The GCF of the variable parts \( x^3 \), \( x^2 \), and \( x \) is \( x \).

Factor out \( 2x \):

$$ 2x(49x^2 + 70x + 25) $$

Step 2: Factor the remaining trinomial

Examine the trinomial inside the parentheses: \( 49x^2 + 70x + 25 \).

This is a perfect square trinomial of the form \( a^2 + 2ab + b^2 = (a + b)^2 \):

• \( 49x^2 = (7x)^2 \), so \( a = 7x \)
• \( 25 = 5^2 \), so \( b = 5 \)
• The middle term is \( 2ab = 2(7x)(5) = 70x \)

Factor the perfect square trinomial:

$$ 49x^2 + 70x + 25 = (7x + 5)^2 $$

Combine this with the GCF factored out in Step 1:

$$ 2x(7x + 5)^2 $$

Answer:

A. \( 98x^3 + 140x^2 + 50x = 2x(7x + 5)^2 \)