Problem 98 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ y^{2}+9-6 y-4 x^{2} $$

Step-by-Step Solution

Answer

The polynomial factors to \((y-3 + 2x)(y-3 - 2x)\).

Step 1: Rearrange the Polynomial

First, rearrange the terms of the polynomial in a standard form. The given polynomial is \( y^2 + 9 - 6y - 4x^2 \). We can write it as \( y^2 - 6y + 9 - 4x^2 \).

Step 2: Recognize the Perfect Square

Notice that the first three terms \( y^2 - 6y + 9 \) can be rewritten as a perfect square. Specifically, \( y^2 - 6y + 9 = (y-3)^2 \).

Step 3: Recognize the Difference of Squares

Now recognize that the expression \( (y-3)^2 - 4x^2 \) is a difference of squares. The formula for the difference of squares is \( a^2 - b^2 = (a + b)(a - b) \).

Step 4: Apply the Difference of Squares Formula

In the expression \( (y-3)^2 - 4x^2 \), let \( a = (y-3) \) and \( b = 2x \). Applying the difference of squares formula, we get: \( ((y-3) + 2x)((y-3) - 2x) \).

Step 5: Write the Factored Form

Thus, the polynomial \( y^2 + 9 - 6y - 4x^2 \) factors to \( (y-3 + 2x)(y-3 - 2x) \). This is the factored form of the given polynomial.

Key Concepts

Difference of Squares

The difference of squares is a special algebraic identity that allows us to factor certain types of expressions. Consider an expression of the form \( a^2 - b^2 \). This expression can be rearranged into \( (a + b)(a - b) \). This is extremely useful for factoring polynomials because it simplifies the process considerably.

To identify a difference of squares, look for terms that are perfect squares themselves separated by a subtraction sign. For instance, if you see \( (y-3)^2 - 4x^2 \), it's a prime candidate for the difference of squares because both \( (y-3)^2 \) and \( 4x^2 \) are perfect squares.

• Formula: \( a^2 - b^2 = (a + b)(a - b) \)
• Example: \( (y-3)^2 - 4x^2 \) factors into \( (y-3 + 2x)(y-3 - 2x) \)

Recognizing this pattern can help you quickly factor expressions in exercises and problems.

Perfect Square Trinomial

Perfect square trinomials are another important concept in algebra that often appears in polynomial factoring. A perfect square trinomial is an expression of the form \( a^2 \pm 2ab + b^2 \), which can be factored into \( (a \pm b)^2 \).

For example, consider the expression \( y^2 - 6y + 9 \). This is a classic case of a perfect square trinomial because it can be rewritten as \( (y-3)^2 \). The method involves recognizing that the middle term, \(-6y\), is twice the product of \( y \) and \( -3 \), making it fit perfectly into the formula of \( (a \pm b)^2 \).

Some tips on recognizing a perfect square trinomial:

• Check if the first and last terms are perfect squares.
• Verify if the middle term is \( \pm 2ab \).
• If both conditions are met, it's a perfect square trinomial.

Understanding and identifying these can greatly simplify the polynomial factoring process.

Algebraic Expressions

Algebraic expressions involve variables, numbers, and operations that come together to form an expression. When working with polynomials, an algebraic expression might include terms with various degrees and coefficients.

Factorization of these expressions is a fundamental skill in algebra. It involves breaking down a complex expression into simpler, multiplied terms (factors) that, when multiplied together, give the original expression.

For online learning, it's crucial to practice the following:

• Identify: Recognize patterns like difference of squares and perfect square trinomials.
• Rearrange: Ensure the polynomial is in a structured format, often descending order by degree.
• Apply: Use factoring formulas to simplify the expression.

Algebraic expressions appear in almost all areas of mathematics, making these foundational skills very beneficial for any student.