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Chapter 6: Problem 21

Use the guess and check method to factor. Identify any prime polynomials. $$ 5 m^{2}-2 m-3 $$

Short Answer

Expert verified
5m^2 - 2m - 3 factors to (5m + 3)(m - 1).

Step by step solution

01

- Set Up the Equation for Factoring

Start with the quadratic equation given: 5m^2 - 2m - 3 We need to factor it in the form (am + b)(cm + d) .
02

- Identify Possible Factors

Given that the quadratic term is 5m^2, the possible pairs for a and c are (5, 1) or (1, 5). For the constant term (-3), the possible pairs for b and d are (-3,1), (3,-1), (1,-3), or (-1,3).
03

- Multiply and Check

Combine the pairs and check each combination by multiplying them out to see which pair gives the middle term of -2m. For instance: Attempt 1: (5m + 3)(1m - 1) = 5m^2 - 5m + 3m - 3 = 5m^2 - 2m - 3 This combination works because the middle term adds up to -2m and it fits the equation.

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Key Concepts

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

Guess and Check Method
The guess and check method is a straightforward strategy for factoring quadratic equations. It involves testing various combinations of factors to find the right pair that satisfies the original equation.

Here's a step-by-step approach to using the guess and check method:
  • Start by writing down the factor pairs for the coefficients of the quadratic and constant terms.
  • Combine these pairs and multiply them to see which pair results in the original equation.
  • Continue testing different pairs until you find the correct factorization.
This method can be time-consuming but is highly effective with practice.
Quadratic Equations
Quadratic equations take the general form:
\( ax^2 + bx + c = 0 \)

They involve a term with a variable squared (second power). In our example, we have: \( 5m^2 - 2m - 3 \).

The goal of factoring a quadratic equation is to rewrite it as a product of two binomials: \( (am + b)(cm + d) \). This makes it easier to solve since setting each factor to zero allows you to find the roots of the equation. Quadratic equations often appear in physics, engineering, and many other fields, making it a crucial concept to understand.
Prime Polynomials
In polynomial factorization, some polynomials cannot be factored into the product of two binomials with real coefficients. These are known as prime polynomials.

A quadratic polynomial is prime if it cannot be factored further, just like prime numbers in arithmetic.

When using the guess and check method, if none of the combinations of factors work to rewrite the polynomial into two binomials, the polynomial is considered prime. It's essential to recognize this to avoid unnecessary calculations.
Polynomial Factorization
Polynomial factorization is the process of expressing a polynomial as a product of its factors.

For quadratic polynomials, we often look to factorize in the form \( (am + b)(cm + d) \).

Steps include:
  • Identifying pairs of factors for the coefficients of the quadratic and constant terms.
  • Combining these pairs to test if they multiply back to the original polynomial.
  • Using techniques like the guess and check method to identify the correct pair of factors.
This process simplifies solving the quadratic equation and can reveal valuable information about its roots.

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Most popular questions from this chapter

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