Problem 1 - A Graphical Approach to College Algebra

Question

Match each polynomial in Column I with its factored form in Column II. I (a) \(x^{2}+10 x y+25 y^{2}\) (b) \(x^{2}-10 x y+25 y^{2}\) (c) \(x^{2}-25 y^{2}\) (d) \(25 y^{2}-x^{2}\) II A. \((x+5 y)(x-5 y)\) B. \((x+5 y)^{2}\) C. \((x-5 y)^{2}\) D. \((5 y+x)(5 y-x)\)

Step-by-Step Solution

Answer

(a) - B, (b) - C, (c) - A, (d) - D.

Step 1: Identify Factoring Patterns

Polynomials can often be factored using special factoring formulas such as perfect square trinomials and difference of squares. Perfect square trinomials follow the pattern \( a^2 \pm 2ab + b^2 = (a \pm b)^2 \), while the difference of squares uses \( a^2 - b^2 = (a + b)(a - b) \).

Step 2: Factor Polynomial (a)

The polynomial \( x^{2}+10xy+25y^{2} \) matches the pattern of a perfect square trinomial, specifically \( (a + b)^2 \). Here, \( a = x \) and \( b = 5y \), as \( 2ab = 10xy \). Therefore, it factors into \((x+5y)^2\).

Step 3: Factor Polynomial (b)

The polynomial \( x^{2}-10xy+25y^{2} \) is another perfect square trinomial, matching the pattern \( (a - b)^2 \). In this case, \( a = x \) and \( b = 5y \), as \( -2ab = -10xy \). It factors into \((x-5y)^2\).

Step 4: Factor Polynomial (c)

The polynomial \( x^{2}-25y^{2} \) fits the difference of squares pattern \( a^2 - b^2 \), with \( a = x \) and \( b = 5y \). This factors into \((x+5y)(x-5y)\).

Step 5: Factor Polynomial (d)

The polynomial \( 25y^{2}-x^{2} \) is a difference of squares as well, written in reverse order. It can be expressed as \((5y)^2-x^2\), which factors into \((5y+x)(5y-x)\).

Step 6: Match Columns

Using the factored forms from previous steps:- (a) matches with II: B, because (a) factors into \((x+5y)^2\).- (b) matches with II: C, because (b) factors into \((x-5y)^2\).- (c) matches with II: A, because (c) factors into \((x+5y)(x-5y)\).- (d) matches with II: D, because (d) factors into \((5y+x)(5y-x)\).

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are a special type of polynomial that follows a specific pattern, making them straightforward to factor. They are usually of the form \( a^2 \pm 2ab + b^2 \). Here's what this means:

• The expression \( a^2 \) represents the square of term \( a \).
• \( 2ab \) is twice the product of two terms, \( a \) and \( b \).
• \( b^2 \) is the square of the term \( b \).

When you see a trinomial that fits this structure, you can factor it directly using one of these patterns:

• If the middle term is positive, as in \( a^2 + 2ab + b^2 \), it can be factored as \((a+b)^2\).
• If the middle term is negative, such as \( a^2 - 2ab + b^2 \), it factors into \((a-b)^2\).

Applying these patterns allows you to transform complex-looking polynomials into simpler factored forms, making them easier to interpret and work with.
In our exercise, polynomials like \( x^2 + 10xy + 25y^2 \) become easily factored because it matches \( (a+b)^2 \), with \( a = x \) and \( b = 5y \), leading to \((x+5y)^2\).

Difference of Squares

The difference of squares is another common pattern in polynomials, recognizable by its simple structure \( a^2 - b^2 \). This pattern is defined by two essential characteristics:

• Two terms that are squares.
• A subtraction operation between these two squared terms.

Using the difference of squares property, the polynomial can be factored as:

• \( a^2 - b^2 = (a + b)(a - b) \)

This tells us you can break it down into two separate binomials – one with addition and the other with subtraction between the squared terms.
For instance, with \( x^2 - 25y^2 \), notice that \( x^2 \) is a square, and \( 25y^2 \) is also a square \((5y)^2\). Hence, using the difference of squares, it simplifies to \((x+5y)(x-5y)\).
This method simplifies solving polynomial equations significantly, as it breaks them into manageable parts.

Factoring Techniques

Factoring techniques are crucial in breaking down polynomials into simpler components. They are methods used to express a polynomial as a product of its factors. Sometimes these techniques involve special formulas such as perfect square trinomials and differences of squares. Here's why these methods are important:

• They simplify complex polynomial expressions.
• Help in solving polynomial equations by making it easier to find zeroes or roots.
• Assist in graphical representation by breaking polynomials into scalable parts.

To apply these techniques effectively, recognizing patterns is essential:

• Identity special forms like perfect squares and difference of squares immediately.
• Look for common factors in each term.
• Write polynomials in descending power order to ease factorization.

In the context of our exercise, identifying the type of polynomial pattern (like the terms seen in \( x^2 - 25y^2 \) or \( x^2 + 10xy + 25y^2 \)) instantly guides the factoring method, ensuring accuracy and efficiency. Mastering these techniques will enhance both your problem-solving skills and your mathematical understanding overall.