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Chapter 5: Problem 23

Factor each trinomial. \(m^{2}-11 m+60\)

Short Answer

Expert verified
The factored form of \(m^2 - 11m + 60\) is \((m-5)(m-6)\).

Step by step solution

01

- Identify the coefficients

First, write down the quadratic trinomial in standard form: . Here, the coefficient of is , the coefficient of is , and the constant term is .
02

- Find two numbers that multiply to the constant term (c) and add to the coefficient of the middle term (b)

We need to find two numbers that multiply to (the constant term, c) and add to (the coefficient of the middle term, b). . The pair of numbers that meet these conditions is and , because and .
03

- Write the trinomial as a product of two binomials

Using the numbers found in Step 2, we can express the original trinomial as a product of two binomials: The trinomial \(m^2 - 11m + 60\) can be factored as \((m-5)(m-6)\).

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Key Concepts

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

quadratic trinomials
A quadratic trinomial is a polynomial expression of degree 2 with three terms. The general form is given by \[ ax^2 + bx + c \]. Here, 'a,' 'b,' and 'c' are coefficients, and each term represents different parts of the polynomial.
Quadratic trinomials appear frequently in algebra, and factoring them is an essential skill. Factoring involves breaking down the trinomial into simpler expressions, known as binomials.
This helps solve quadratic equations more easily. It's crucial to practice recognizing patterns and applying strategies to factor quadratic trinomials effectively.
binomials
A binomial is a polynomial expression with exactly two terms, usually written in the form \[ (x + m)(x + n) \] or \[ (x - m)(x - n) \]. Binomials are significant because they are often the factors of quadratic trinomials.
To factor a quadratic trinomial like \[ m^2 - 11m + 60 \], you need to express it as a product of two binomials.
For this example, the quadratic trinomial \[ m^2 - 11m + 60 \] is factored as \[ (m - 5)(m - 6) \].
Binomials simplify complex expressions, making it easier to solve equations and understand their structure.
coefficients
Coefficients are numerical or constant factors multiplying the variables in a polynomial expression. In a quadratic trinomial like \[ m^2 - 11m + 60 \]:
  • The coefficient of \[ m^2 \] is 1.
  • The coefficient of \[ m \] is -11.
  • The constant term is 60.

Understanding coefficients is critical in solving and factoring trinomials. They help identify the relationships between the terms and enable you to find the suitable binomials that multiply to form the original expression. For example, in \[ m^2 - 11m + 60 \]: the coefficients guide us to the correct factors, \[ (m - 5) \] and \[ (m - 6) \], because \[ (-5) + (-6) = -11 \] and \[ (-5) * (-6) = 30 \] .

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