Question

1. factor the following polynomial completely. if it cannot be factored, write prime polynomial.

$t^2 + 196$

1. factor the following perfect square polynomial. it can be factored, so do not write prime polynomial.

$9x^2 - 30xy + 25y^2$

Explanation:

Problem 4: Factor \( t^2 + 196 \)

Step 1: Recall factoring rules

The polynomial is a sum of squares: \( t^2 + 14^2 \). In real numbers, \( a^2 + b^2 \) doesn't factor (it's prime), since \( a^2 + b^2=(a + bi)(a - bi) \) in complex numbers, but over reals, no linear factors.

Step 1: Identify perfect square form

A perfect square trinomial has the form \( a^2 - 2ab + b^2=(a - b)^2 \). Here, \( 9x^2=(3x)^2 \), \( 25y^2=(5y)^2 \), and \( -30xy=-2(3x)(5y) \).

Step 2: Apply perfect square formula

Using \( a = 3x \), \( b = 5y \), we get \( (3x - 5y)^2 \).

Answer:

Prime (since \( t^2 + 196 \) is a sum of squares and doesn't factor over real numbers)

Problem 5: Factor \( 9x^2 - 30xy + 25y^2 \)