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Chapter 6: Problem 42

For Problems \(11-46\), factor each polynomial completely. (Objective 2) $$ x(y-6)-3(y-6) $$

Short Answer

Expert verified
The factored form is \((y-6)(x-3)\).

Step by step solution

01

Identify Common Factors

The expression is given as \( x(y-6) - 3(y-6) \). Notice that \((y-6)\) is a common factor in both terms.
02

Factor Out the Common Term

Since \((y-6)\) is common, we factor it out from both terms. This gives: \((y-6)(x-3)\).
03

Verify the Factored Form

To ensure accuracy, expand \((y-6)(x-3)\) back to its original form. Expanding gives \(x(y-6) - 3(y-6)\), which matches the original polynomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

common factor
When dealing with polynomials, a common factor is an expression that appears in all terms of a given polynomial. Finding a common factor simplifies the polynomial and is an essential step in factoring completely.
Imagine you have multiple terms, such as the expression \( x(y-6) - 3(y-6) \).
Both terms in this example, \( x(y-6) \) and \( -3(y-6) \), clearly share the factor \( (y-6) \).
Identifying this common factor allows for the simplification of the expression.
  • This involves looking at all terms and spotting the identical factor that is repeated.
  • The goal is to "pull out" or "factor out" this shared element, much like finding the greatest common factor in arithmetic.
Spotting the common factor early on makes solving polynomial problems much more straightforward.
It reduces the polynomial to a simpler form, making it easier to work with in further calculations.
polynomial expression
A polynomial expression is a mathematics term used to describe an algebraic expression involving variables raised to various powers (exponents) along with coefficients.
These expressions can involve operations like addition, subtraction, and multiplication.
Polynomials are crucial because they represent a wide range of algebraic relationships and functions.
  • Think of polynomial expressions as a sum of terms, each involving a variable raised to a non-negative integer power.
  • For instance, the expression \( x(y-6) - 3(y-6) \) is a polynomial in terms of the variable \( y \).
They provide a way to understand the relationships and changes between different variables.
In solving polynomial problems, breaking down these expressions by finding common factors or rearranging through operations is often required.
expand and verify polynomials
Expanding and verifying are critical concepts when working with polynomials, ensuring that the factored form is accurate.
Expansion is the process of multiplying factors back out to return to the original polynomial.
Verification involves confirming that the expansion results in the initial expression.
  • To expand a polynomial like \((y-6)(x-3)\), apply the distributive property: Multiply each term by each other to check correctness.
  • This results in \( x(y-6) - 3(y-6) \), which is the original expression, confirming the accuracy of the factorization.
This step of expanding and verifying is crucial because it prevents mistakes and ensures that the complete factoring process is correct.
By re-expanding, you safeguard against errors and can confidently rely on your solution.

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Most popular questions from this chapter

\(x^{2}-9 x+8=0\)

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ (2 n+5)(n+4)=-1 $$

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) The sum of the areas of two circles is \(100 \pi\) square centimeters. The length of a radius of the larger circle is 2 centimeters more than the length of a radius of the smaller circle. Find the length of a radius of each circle.

\(x^{2}+11 x=12\)

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ 2 t^{3}-16 t^{2}-18 t=0 $$
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