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A perfect square trinomial is a special type of quadratic expression. It takes the form \(a^2 + 2ab + b^2\) and can be factored into \((a + b)^2\). For example, in the expression \(64x^2 + 16xy + y^2\), observe the components:

• The term \(64x^2\) is a perfect square because it equals \((8x)^2\).
• The term \(y^2\) is also a perfect square, since it equals \(y^2\).
• The term \(16xy\) is twice the product of \(8x\) and \(y\), or \(2(8x)(y)\).

Thus, this trinomial can be expressed as \((8x + y)^2\). Recognizing these patterns can make factoring perfect square trinomials much easier!

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators. In our context, the given expression \(64x^2 + 16xy + y^2\) is an algebraic expression consisting of terms that involve variables \(x\) and \(y\).
Let's break down the components:

• \(64x^2\): This is a term with the variable \(x\) raised to the second power, multiplied by 64.
• \(16xy\): This term involves the product of two different variables, \(x\) and \(y\), multiplied by 16.
• \(y^2\): This term involves the variable \(y\) raised to the second power.

Understanding algebraic expressions is crucial when working with polynomials and factoring them. By getting comfortable with how terms, coefficients, and variables operate, you'll find it easier to manipulate and simplify these expressions.

Polynomial factorization involves breaking down a polynomial into simpler components, or 'factors,' that when multiplied together give you the original polynomial.
For our given expression \(64x^2 + 16xy + y^2\), factorization converts it into a product of simpler polynomials. Recognizing that it's a perfect square trinomial is the key here.
The given expression can be rewritten using the perfect square trinomial formula:
\[ (8x + y)^2 \] This tells us that the polynomial \(64x^2 + 16xy + y^2\) factors into \((8x + y)(8x + y)\). Proper factorization makes solving equations easier and is a fundamental skill in algebra. Practice breaking down trinomials and other polynomials into their factors to get more comfortable with this process!