Problem 84 - Introductory Algebra for College Students

Question

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$x y-5 x+2 y-10$$

Step-by-Step Solution

Answer

After complete factorization the polynomial \(xy - 5x + 2y - 10\) becomes \( (x + 2)(y - 5)\).

Step 1: Rearrange the polynomial

In order to factorizing well, let's rearrange the polynomial like this: \(x y - 5x + 2y - 10 = x(y - 5) + 2(y - 5)\)

Step 2: Factor out common terms

Now, factor out the common terms from the polynomial: \[x (y - 5) + 2 (y - 5) = (x + 2)(y - 5)\]

Step 3: Verification through multiplication

The factoring can be rechecked via multiplication. We multiply the factored form again to verify if it gives back the original polynomial.\[ (x + 2)(y - 5)= x y - 5x + 2y - 10 \]

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. They can have one or more terms.
For example, in the expression \(xy - 5x + 2y - 10\), we have a polynomial with four terms. These terms involve the variables \(x\) and \(y\), constants, and coefficients that accompany each term.
Polynomials are fundamental in algebra and can be operated on in various ways, including addition, subtraction, multiplication, and factoring.
Understanding how to manipulate these expressions is crucial in solving equations and simplifying complex expressions throughout algebra and calculus.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials that involves rearranging and grouping terms to find common factors. This method works best when you cannot directly factor the polynomial using other simpler methods.
The process typically involves:

• Grouping the terms with common factors.
• Factoring out the common factor from each group.
• Looking for a common binomial factor to factor out from the new expression.

In our exercise, we had the polynomial \(xy - 5x + 2y - 10\).
We rearranged it to group similar terms: \((xy - 5x) + (2y - 10)\), then factored by grouping to reveal \((x + 2)(y - 5)\).
This new factorization simplifies our polynomial into a form that is often easier to work with and solves problems more efficiently.

Verification by Multiplication

Verification by multiplication ensures that your factoring is correct. By multiplying your factored polynomial, you should be able to recreate the original polynomial.
This step is crucial because it confirms accuracy and helps avoid mistakes in subsequent calculations.
For the factored expression \((x+2)(y-5)\) from our exercise, multiplication involves expanding the expression using the distributive property:

• Multiply \((x + 2)\) by \(y - 5\).
• Your result should give \(xy - 5x + 2y - 10\).

This process is essentially reversing the factoring steps and is a reliable way to check the solution. When multiplied correctly, the original polynomial should be restored, proving the factorization was done accurately.