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Chapter 1: Problem 107

Factor the expression completely. $$4 x^{2}+4 x y+y^{2}$$

Short Answer

Expert verified
The expression factors to \((2x + y)^2\).

Step by step solution

01

Identify the expression structure

Recognize that the expression is a quadratic trinomial of the form \[ ax^2 + bxy + cy^2 \], where \( a = 4 \), \( b = 4 \), and \( c = 1 \). This suggests a perfect square trinomial, which has the form \((px + qy)^2\).
02

Check for perfect square trinomial

A perfect square trinomial has the form \[ (px + qy)^2 = p^2x^2 + 2pqxy + q^2y^2 \].For our expression, compare coefficients:1. \( a = p^2 = 4 \) which gives \( p = 2 \).2. \( 2pq = 4 \) plugging in \( p = 2 \) gives \( q = 1 \).3. \( c = q^2 = 1 \) checks out with \( q = 1 \).
03

Factor using the perfect square formula

Since all conditions for a perfect square trinomial are met, factor the expression as \[ (2x + y)^2 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Trinomial
A perfect square trinomial is a special type of expression that can be written as the square of a binomial. The form it typically takes is \( (px + qy)^2 \). This means the expression can be expanded into \( p^2x^2 + 2pqxy + q^2y^2 \).

To check if an expression is a perfect square trinomial, one must look for a pattern:
  • Check if the first and last terms are perfect squares.
  • Examine if the middle term is twice the product of the square roots of the first and last terms.
These characteristics help in quickly recognizing and factoring the expression back to its binomial square.

In our original problem, recognizing the structure \( 4x^2 + 4xy + y^2 \) as \( p^2x^2 + 2pqxy + q^2y^2 \) enables us to apply this method and factor the expression as \( (2x + y)^2 \).
Quadratic Expression
Quadratic expressions are polynomials of the second degree. These expressions can be written in the general form \( ax^2 + bxy + cy^2 \) when dealing with two variables. The 'quadratic' signifies that the highest degree of the variable in the expression is two.

Understanding the role of each term:
  • The \( ax^2 \) term is known as the quadratic term.
  • The \( bxy \) term is the interaction between \( x \) and \( y \).
  • The \( cy^2 \) is the constant or constant term.
Recognizing these components helps in understanding the underlying structure of more complex algebraic expressions.

In the original exercise, the expression \( 4x^2 + 4xy + y^2 \) is a classic quadratic where the distinct variables play crucial roles in revealing the expression's true form.
Coefficient Comparison
Coefficient comparison is a crucial technique in algebra, particularly when working with quadratic expressions to identify structural patterns. This involves analyzing and matching the coefficients from a given expression to a known form.

For the expression \( ax^2 + bxy + cy^2 \), consider:
  • The \( a = p^2 \), which gives the coefficient of the \( x^2 \) term.
  • The \( 2pq = b \), the coefficient of the \( xy \) term.
  • Lastly, \( c = q^2 \), providing the coefficient of the \( y^2 \) term.
By comparing these coefficients, one can determine the values of \( p \) and \( q \), effectively revealing if the expression is a perfect square trinomial.

In the step-by-step solution provided, this method was applied to determine \( p = 2 \) and \( q = 1 \), thus supporting the factorization of the expression \( 4x^2 + 4xy + y^2 \) as \( (2x + y)^2 \).

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Most popular questions from this chapter

Factor the expression completely. $$x^{2}+10 x+25$$

Use a Special Factoring Formula to factor the expression. $$a^{3}-b^{6}$$

Verify these formulas by expanding and simplifying the right-hand side. $$ \begin{array}{l} A^{2}-1=(A-1)(A+1) \\ A^{3}-1=(A-1)\left(A^{2}+A+1\right) \\ A^{4}-1=(A-1)\left(A^{3}+A^{2}+A+1\right) \end{array} $$ On the basis of the pattern displayed in this list, how do you think \(A^{5}-1\) would factor? Verify your conjecture. Now generalize the pattern you have observed to obtain a factoring formula for \(A^{n}-1,\) where \(n\) is a positive integer.

Factor the expression by grouping terms. x^{3}+4 x^{2}+x+4

Factor the expression completely. $$18 y^{3} x^{2}-2 x y^{4}$$
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