Problem 71 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 49 p^{2}+28 p q+4 q^{2} $$

Step-by-Step Solution

Answer

The expression factors to \((7p + 2q)^2\).

Step 1: Identify the form of the expression

The expression is a quadratic trinomial of the form \(Ax^2 + Bxy + Cy^2\). We are given \(49p^2 + 28pq + 4q^2\).

Step 2: Check if it is a perfect square trinomial

Check if the expression is a perfect square trinomial by matching it against the pattern \((mx + ny)^2 = m^2x^2 + 2mnxy + n^2y^2\) where \(m^2 = A\), \(n^2 = C\), and \(2mn = B\).

Step 3: Evaluate components of the perfect square

For \(A = 49\), we have \(m^2 = 49\) implying \(m = 7\). For \(C = 4\), we have \(n^2 = 4\) implying \(n = 2\). To match \(B = 28\) with \(2mn\), we compute \(2 \times 7 \times 2 = 28\).

Step 4: Rewrite the expression as a squared binomial

Since the expression satisfies the perfect square trinomial conditions, we can write it as \((7p + 2q)^2\).

Step 5: Conclusion on factorization

The expression \(49p^2 + 28pq + 4q^2\) is factored as \((7p + 2q)^2\), which confirms it is a perfect square trinomial.

Key Concepts

Perfect Square Trinomials

Understanding perfect square trinomials can be quite rewarding, as they simplify into a recognizable pattern. A perfect square trinomial is a specific kind of quadratic trinomial. These are expressions that can be rewritten as the square of a binomial. The general formula to spot a perfect square trinomial is:

• \(A = m^2\)
• \(B = 2mn\)
• \(C = n^2\)

So, when you have an expression of the form \(Ax^2 + Bxy + Cy^2\), you should check if it matches the format \( (mx + ny)^2 = m^2x^2 + 2mnxy + n^2y^2\). Identifying the components correctly (\(m\) and \(n\) here) helps in determining the binomial whose square equals the given trinomial. This requires understanding the square of a binomial is expanded to yield a trinomial format. Recognizing perfect square trinomials enables efficient factorization, ultimately simplifying solving equations or further algebraic manipulations.

Algebraic Expressions

Algebraic expressions are a fundamental concept in algebra. They consist of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. An algebraic expression like \(49p^2 + 28pq + 4q^2\) involves:

• Coefficients, like 49 and 4
• Variables, here \(p\) and \(q\)
• Exponents, as seen in \(p^2\) and \(q^2\)
• Terms, which are the separated parts of the addition or subtraction, such as \(49p^2\), \(28pq\) and \(4q^2\)

Algebraic expressions can be manipulated in various steps to solve problems. By familiarizing oneself with writing, interpreting, and manipulating these expressions, students gain skills to approach complex problems logically. Recognizing patterns in algebraic expressions can lead to efficient problem-solving strategies and thus ease the application of mathematical concepts across different areas.

Mathematical Factorization

Mathematical factorization is the process of breaking down an expression into a product of simpler expressions or factors. In the case of factoring quadratic trinomials like \(49p^2 + 28pq + 4q^2\), factorization involves finding a way to express it as a product of binomials.

• Identify the structure of the polynomial: Check whether it's a quadratic, cubic, or higher degree polynomial.
• Determine patterns within the terms: Look for perfect squares, sums, or products that match known factorization patterns.
• Rewrite the expression into simpler multiplied terms: For perfect square trinomials, reduce them to the square of a binomial (e.g., \((7p + 2q)^2\)).

Factorization simplifies problems, reduces equations, and helps solve complicated algebraic tasks. It is a crucial skill for higher mathematics, making it easier to simplify expressions and solve equations systematically. For students, mastering factorization opens doors to diverse problem-solving techniques and insights.