Question

describing a polynomial

which statements are true about the polynomial \\(28vw + 49v + 35w\\)? check all that apply.

the coefficients have no common factors other than 1.
there are no common variables among all three terms.
the gcf of the polynomial is \\(7v\\).
each term written as the product, where one factor is the gcf, is \\(7(28vw) + 7(49v) + 35(w)\\).
the resulting expression when factoring out the gcf is \\(7(4vw + 7v + 5w)\\).

Explanation:

Find the GCF of the coefficients

We find the greatest common factor of the numerical coefficients \(28\), \(49\), and \(35\).
Using the Greatest Common Factor knowledge point:

• Factors of \(28\): \(1, 2, 4, 7, 14, 28\)
• Factors of \(49\): \(1, 7, 49\)
• Factors of \(35\): \(1, 5, 7, 35\)

The greatest common numerical factor is \(7\).

Find the GCF of the variables

We analyze the variable parts of the three terms: \(vw\), \(v\), and \(w\).
Using the Greatest Common Factor of Monomials knowledge point:

• The first term contains \(v\) and \(w\).
• The second term contains only \(v\).
• The third term contains only \(w\).

There is no variable common to all three terms. Thus, the variable GCF is \(1\).

Determine the overall GCF of the polynomial

We combine the numerical and variable factors.
Using the Greatest Common Factor knowledge point:

$$ \text{GCF} = 7 \times 1 = 7 $$

Evaluate each statement

We test each given option:

1. "The coefficients have no common factors other than 1."

• False, because they share a common factor of \(7\).

1. "There are no common variables among all three terms."

• True, as shown in our variable analysis.

1. "The GCF of the polynomial is \(7v\)."

• False, the GCF is \(7\).

1. "Each term written as the product, where one factor is the GCF, is \(7(28vw) + 7(49v) + 35(w)\)."

• False, writing each term with a factor of \(7\) would be \(7(4vw) + 7(7v) + 7(5w)\).

1. "The resulting expression when factoring out the GCF is \(7(4vw + 7v + 5w)\)."

• True, factoring out \(7\) from \(28vw + 49v + 35w\) yields:

Using the Factoring Polynomials knowledge point:

$$ 7(4vw + 7v + 5w) $$

Answer:

• The coefficients have no common factors other than 1.
• There are no common variables among all three terms. (Correct answer)
• The GCF of the polynomial is 7v.
• Each term written as the product, where one factor is the GCF, is 7(28vw) + 7(49v) + 35(w).
• The resulting expression when factoring out the GCF is 7(4vw + 7v + 5w). (Correct answer)