Question

what is the polynomial \\(3y^2 + (y + 7)^2 - 15\\) after it has been fully simplified and written in standard form?

\\(4y^2 + 34\\)

\\(4y^4 + 34\\)

\\(4y^2 + 14y + 34\\)

\\(3y^2 + y + 34\\)

Explanation:

Expand the squared binomial

Using the Polynomial Multiplication knowledge point

$$ (y + 7)^2 = (y + 7)(y + 7) = y^2 + 14y + 49 $$

Substitute and combine like terms

We substitute the expanded binomial back into the original expression:

$$ 3y^2 + (y^2 + 14y + 49) - 15 $$

Now, we group and combine the like terms:

• Quadratic terms: \(3y^2 + y^2 = 4y^2\)
• Linear terms: \(14y\)
• Constant terms: \(49 - 15 = 34\)

Write in standard form

A polynomial is in standard form when its terms are written in descending order of their degrees (from highest exponent to lowest):

$$ 4y^2 + 14y + 34 $$

This matches the third option.

Answer:

• (A) \(4y^2 + 34\)
• (B) \(4y^4 + 34\)
• (C) \(4y^2 + 14y + 34\) (Correct answer)
• (D) \(3y^2 + y + 34\)