Problem 16 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$x^{2}-14 x y+40 y^{2}$$

Step-by-Step Solution

Answer

The polynomial factors to \((x - 10y)(x - 4y)\).

Step 1: Identify the Polynomial

The polynomial given is a quadratic expression: \(x^2 - 14xy + 40y^2\). It is a trinomial with respect to \(x\) and involves terms with \(y\) as the variable.

Step 2: Factor the Trinomial

To factor the trinomial \(x^2 - 14xy + 40y^2\), we need to find two numbers that multiply to \(40y^2\) (the constant term) and add to \(-14y\) (the coefficient of the middle term). These numbers are \(-10y\) and \(-4y\).

Step 3: Write the Expression as a Product of Binomials

Using the numbers found, the trinomial can be rewritten as a product of two binomials: \((x - 10y)(x - 4y)\).

Step 4: Verify the Factorization

To ensure correctness, expand \((x - 10y)(x - 4y)\): - \(x \cdot x = x^2\)- \(x \cdot -4y = -4xy\)- \(-10y \cdot x = -10xy\)- \(-10y \cdot -4y = 40y^2\)Combine these terms: \(x^2 - 14xy + 40y^2\), which confirms that the factorization is correct.

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial where the highest degree of the variable is two. These are commonly represented in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is a variable. In our example, the quadratic expression is \( x^2 - 14xy + 40y^2 \). Here, \( x \) is the primary variable, but it also includes terms with another variable \( y \). This adds complexity as it involves cross terms like \( xy \).Quadratic expressions are vital in algebra as they lay the foundation for more advanced topics, such as quadratic equations and functions. Understanding how to manage these expressions is essential for solving various mathematical problems efficiently.

Trinomial Factoring

Trinomial factoring involves breaking down a three-term polynomial into the product of two binomials. This is a common technique used to simplify expressions and solve equations.In the expression \( x^2 - 14xy + 40y^2 \), consider it:

• The coefficient of \( x^2 \) is 1.
• We need to find two numbers that multiply to the product of the first and last coefficients, here that’s \( 40y^2 \), and add up to the middle coefficient, \( -14y \).

These numbers are \(-10y\) and \(-4y\), as they satisfy both conditions. Using trial, error, and knowledge of multiplication facts, these numbers can be identified. Factoring requires practice, as recognizing patterns is key to simplifying complex expressions.

Binomial Product

Once the appropriate numbers for a trinomial are identified, they can be used to express the polynomial as the product of two binomials. In this case, \( x^2 - 14xy + 40y^2 \) was factored into \((x - 10y)(x - 4y)\).A binomial is a polynomial with two terms, and when two binomials are multiplied together, they produce a trinomial if performed correctly. This multiplication is commonly executed through the distributive property, often referred to as "FOIL" (First, Outside, Inside, Last) in algebra.The calculated product of \( (x - 10y) \) and \( (x - 4y) \) perfectly reconstructs the original trinomial, demonstrating proper factorization.

Polynomial Verification

Polynomial verification ensures that the factorization of a polynomial is accurate. After factoring \( x^2 - 14xy + 40y^2 \) into \((x - 10y)(x - 4y)\), the next step is to verify it by expansion.Here is the breakdown:

• Multiply the first terms: \(x \cdot x = x^2\).
• Combine the outer terms: \(x \cdot (-4y) = -4xy\).
• The inner terms: \(-10y \cdot x = -10xy\).
• Finally, the last terms: \(-10y \cdot (-4y) = 40y^2\).

When these products are combined: \(x^2 - 4xy - 10xy + 40y^2\), simplify to retrieve the original trinomial \(x^2 - 14xy + 40y^2\). This confirms that our factorization was correct, illustrating the importance of verification in algebra to avoid errors.