Problem 36 - Elementary Algebra

Question

Factor completely. $$ 16 x 2-66 x y-27 y 2 $$

Step-by-Step Solution

Answer

The expression factors as \((8x + 3y)(2x - 9y)\).

Step 1: Recognize the Expression Format

The expression given is a quadratic in two variables, where the terms are: \[ 16x^2 - 66xy - 27y^2 \]It fits the general format of a quadratic trinomial \( ax^2 + bxy + cy^2 \).

Step 2: Find Two Numbers with Specific Properties

We need two numbers that multiply to \( a \times c = 16 \times (-27) = -432 \) and add up to \( b = -66 \). These numbers are -72 and 6 because:- \(-72 \times 6 = -432\)- \(-72 + 6 = -66\)

Step 3: Decompose the Middle Term

Rewrite the expression by splitting the middle term \(-66xy\) into \(-72xy + 6xy\):\[ 16x^2 - 72xy + 6xy - 27y^2 \]

Step 4: Factor by Grouping

Group the terms in pairs:1. \( (16x^2 - 72xy) + (6xy - 27y^2) \)2. Factor out the greatest common factor (GCF) from each pair. - From the first pair \( 16x^2 - 72xy \), factor out \( 8x \): \[ 8x(2x - 9y) \] - From the second pair \( 6xy - 27y^2 \), factor out \( 3y \): \[ 3y(2x - 9y) \]

Step 5: Combine with Common Factor

Now the expression is \((8x)(2x - 9y) + (3y)(2x - 9y)\). Notice that \((2x - 9y)\) is a common factor.Thus, factor \((2x - 9y)\) out:\[ (8x + 3y)(2x - 9y) \]

Step 6: Check the Factorization

Verify your factorization by expanding \((8x + 3y)(2x - 9y)\) to see if it gives the original expression: \[ (8x + 3y)(2x - 9y) = 16x^2 - 72xy + 6xy - 27y^2 = 16x^2 - 66xy - 27y^2 \] The factorization is correct.

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials, especially when there are four or more terms. It involves grouping terms into pairs and then factoring out the greatest common factor (GCF) from each pair. Once grouped, it becomes easier to spot common factors that can be factored out of the entire expression.

Here is how you apply factoring by grouping:

• Group the terms into pairs.
• For each pair, identify and factor out the GCF.
• You should notice a common binomial factor after this operation.
• Factor out the common binomial factor from each expression.

For example, if your polynomial is \( ax + ay + bx + by \), you can group it as \((ax + ay) + (bx + by)\). You then factor each group to get \(a(x + y) + b(x + y)\), and finally, you factor out the common term \((x + y)\) to get \((a + b)(x + y)\). In this way, your polynomial is now fully factored.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. In algebra, we often seek the GCF when factoring expressions to simplify them. Finding the GCF is crucial for factoring expressions by grouping.

Here's how you find the GCF of terms:

• Identify the factors of each coefficient.
• Look for the largest factor that is common among all terms.
• If terms have variables, take the variable with the lowest exponent that appears in all terms.

For instance, in \( 16x^2 - 72xy \), the GCF of the coefficients 16 and 72 is 8. Also, both terms have an \( x \), so \( 8x \) is the GCF of the entire expression. By factoring out \( 8x \), we simplify the expression.The GCF is key in reducing expressions into simpler forms and easing further operations like factoring or division.

Two-variable Quadratics

Two-variable quadratics involve expressions with terms having two different variables. These forms are often seen in expressions like \( ax^2 + bxy + cy^2 \). Recognizing this structure is pivotal in understanding how to factor them deeper.

When dealing with these quadratics, the process involves:

• Understanding the expression's layout as a trinomial.
• Finding products and sums that align with the middle term, often through assigned coefficients.
• Employing distribution and factoring methods such as using factoring by grouping.

Considering a quadratic like \( 16x^2 - 66xy - 27y^2 \), we see it follows the trinomial format. Factoring this involves splitting the middle term and grouping the expression to extract the common factors efficiently, leading us to a fully simplified form. Mastery of two-variable quadratics prepares one for more complex algebraic tasks and problem-solving scenarios.