Problem 65 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 45 x^{2}+38 x y+8 y^{2} $$

Step-by-Step Solution

Answer

The polynomial can be factored as \((9x + 4y)(5x + 2y)\).

Step 1: Identify the Quadratic Form

The polynomial given is \( 45x^2 + 38xy + 8y^2 \). It is a quadratic in terms of \( x \) and \( y \), in the general form \( ax^2 + bxy + cy^2 \). Here, \( a = 45 \), \( b = 38 \), and \( c = 8 \).

Step 2: Find Two Binomials

Look for binomials of the form \((px + qy)(rx + sy)\) such that expanding these gives the original polynomial. This means finding \(p, q, r,\) and \(s\) so that \(pr = a = 45\), \(qs = c = 8\), and \(ps + qr = b = 38\).

Step 3: Solve for Coefficients

We will try pairings of factors for 45 and 8. Let's try \( (9x + 4y)(5x + 2y) \). Expanding this results in \( 9x \cdot 5x + 9x \cdot 2y + 4y \cdot 5x + 4y \cdot 2y \). Calculate these products:\[ 9x \cdot 5x = 45x^2, \ensuremath{\quad} 9x \cdot 2y = 18xy, \ensuremath{\quad} 4y \cdot 5x = 20xy, \ensuremath{\quad} 4y \cdot 2y = 8y^2 \] Combining terms gives: \( 45x^2 + 18xy + 20xy + 8y^2 = 45x^2 + 38xy + 8y^2 \).

Step 4: Verify the Factorization

The expanded expression matches the original polynomial \(45x^2 + 38xy + 8y^2\), confirming that the factorization is correct. Therefore, the factorization of the given polynomial is \((9x + 4y)(5x + 2y)\).

Key Concepts

Quadratic Form

Understanding the quadratic form in polynomials helps us recognize patterns that can simplify factoring. A polynomial like \( 45x^2 + 38xy + 8y^2 \) follows the quadratic form. In general, a quadratic in two variables \( x \) and \( y \) can be expressed as \( ax^2 + bxy + cy^2 \). Here, \( a \), \( b \), and \( c \) are coefficients that dictate the relationship between the terms.

To identify this form, look for:

• The presence of an \( x^2 \) term
• The presence of a \( y^2 \) term
• A middle term that includes both \( x \) and \( y \)

Recognizing this structure is the first step in factoring, as it allows us to decide how to approach the problem using methods applicable to quadratics, such as finding patterns for factoring into binomials.

Binomial Products

Binomial products allow us to express a polynomial as a product of two binomials. For example, we want to express \( 45x^2 + 38xy + 8y^2 \) in the form \((px + qy)(rx + sy)\).

Here's what to look for when identifying these factors:

• The product of first terms \( p \, r = a = 45 \)
• The product of last terms \( q \, s = c = 8 \)
• The combined product of outer and inner terms \( p \, s + q \, r = b = 38 \)

By systematically testing factor pairs, such as \((9x + 4y)\) and \((5x + 2y)\), and checking that their expansion matches the original polynomial, we ensure each term is correctly represented.

Verifying each step through multiplication verifies whether the choice fits the original quadratic polynomial, which confirms the binomial factorization is accurate.

Polynomial Expansion

Polynomial expansion involves multiplying binomials to return to the original polynomial. This checks if the factorization is correct. In our example, the factorization \((9x + 4y)(5x + 2y)\) needs verification.

To expand, apply the distributive property:

• \(9x \, \times \, 5x = 45x^2\)
• \(9x \, \times \, 2y = 18xy\)
• \(4y \, \times \, 5x = 20xy\)
• \(4y \, \times \, 2y = 8y^2\)

Combine the terms \(18xy\) and \(20xy\) to get \(38xy\), thus reconstructing the polynomial \(45x^2 + 38xy + 8y^2\).

This final step not only validates our factorization but also deepens our understanding of how binomial expansion correlates with quadratic expressions, confirming the entire process from factorization to expansion is accurate.