Problem 17 - Beginning and Intermediate Algebra

Question

Complete the factorization. $$18 x^{2}-17 x y+4 y^{2}=(2 x-y)(\quad)$$

Step-by-Step Solution

Answer

The complete factorization of the given expression is \((2x-y)(9x-4y)\).

Step 1: Expand the given expression

Expand the given expression and collect like terms: $$ 18x^2 - 17xy + 4y^2 = (2x-y)(Ax+By)$$ Our goal is to determine the values of A and B that make this equation true.

Step 2: Expand and distribute terms

Expand the expression on the right-hand side, distributing the first term in the first bracket by each term in the second bracket and the second term in the first bracket by each term in the second bracket: $$ (2x-y)(Ax+By) = 2x(Ax+By) - y(Ax+By) $$

Step 3: Distribute the terms further

Continue to distribute terms to get the expanded form, which should match the expression on the left: $$ 2x(Ax + By) - y(Ax + By) = (2xAx) + (2xBy) - (yAx) - (yBy) $$ We can rewrite this expression as: $$ 2A x^2 + 2Bxy - Ayx - By^2 $$

Step 4: Compare the coefficients

Now, we compare the coefficients of like terms in both expressions. If the coefficients match, then the expressions are equal: $$ 18x^2 - 17xy + 4y^2 = 2A x^2 + 2B xy - Ayx - By^2$$ Comparing coefficients: - For \(x^2\): \(18 = 2A\) - For \(xy\): \(-17 = 2B - A\) - For \(y^2\): \(4 = -B\)

Step 5: Solve the system of equations

Solve the three equations for A, B, and C: 1. For \(x^2\) coefficients: \(18 = 2A\) => \(A = 9\) 2. For \(y^2\) coefficients: \(4 = -B\) => \(B = -4\) 3. For \(xy\) coefficients: \(-17 = 2B - A\) => \(-17 = 2(-4) - 9\) => \(-17 = -8 - 9\) The values of A and B satisfy all three equations, so we have found the correct factor.

Step 6: Write the complete factorization

Substitute the found values of A and B back into the expression to obtain the complete factorization of the given quadratic expression: $$18x^2 - 17xy + 4y^2 = (2x-y)(9x-4y) $$ So, the complete factorization of the given expression is \((2x-y)(9x-4y)\).

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest power of the variable is two. These are common in algebra and have the general form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In the context of the exercise, we’re dealing with two variables, \( x \) and \( y \), which forms the expression \( 18x^2 - 17xy + 4y^2 \).
Unlike single-variable quadratics which may look like \( ax^2 + bx + c \), this expression involves terms like \( xy \) and \( y^2 \) as well. This requires a strategic approach like factorization.

Factorization involves rewriting the quadratic expression as a product of simpler expressions, often linear in form. For example, our expression \( 18x^2 - 17xy + 4y^2 \) is factorized into the linear terms \((2x-y)(9x-4y)\). Factorization is beneficial because it can simplify equations or simplify the process of finding roots.

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. Solving these equations encompasses finding the values of the variables that make the equation true. In the exercise, we dealt with an equation structured around an equality involving the factorization:

\( 18x^2 - 17xy + 4y^2 = (2x-y)(Ax+By) \)

The task involves expanding and simplifying using distributive properties to identify the values of \(A\) and \(B\) that satisfy the equality. Expanding each part and collecting like terms leads to solving systems of equations derived from comparing coefficients, an approach that showcases how quadratic expressions can be represented algebraically.

• Expanding: Distributing `(2x-y)` with `(Ax+By)` to express it in simplified quadratic terms.
• Comparing: Rearranging terms to match them with the original expression \(18x^2 - 17xy + 4y^2\).

This method showcases solving algebraic equations by converting complex expressions into manageable equations to find unknown coefficients.

Coefficient Comparison

Coefficient comparison is a method used to equate coefficients of corresponding terms on both sides of an equation. This technique is helpful for equations like those presented in polynomial expressions to determine unknown values.

In this problem, the expanded product \((2x-y)(Ax+By)\) results in:

\(2Ax^2 + 2Bxy - Ayx - By^2 \)

These terms are then compared to the original expression \(18x^2 - 17xy + 4y^2\) by setting them equal and matching the coefficients of each term. The procedure follows:

• Match terms with \(x^2\): Set \(18 = 2A\), leading to \(A = 9\).
• Match terms with \(xy\): Set \(-17 = 2B - A\), simplifying to find consistency of values.
• Match terms with \(y^2\): Set \(4 = -B\), solving this reveals \(B = -4\).

By systematically comparing coefficients, we solve for the constants \(A\) and \(B\). This technique exemplifies how polynomial expressions in algebra use coefficient comparison to decipher unknown elements, verifying solutions as consistent within the algebraic structure.