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Chapter 5: Problem 162

Factor each polynomial using the trial-and-error method. $$ 3 x^{2}-25 x+8 $$

Short Answer

Expert verified
(3x - 1)(x - 8)

Step by step solution

01

Identify the coefficients

The coefficients of the quadratic polynomial are: a = 3, b = -25, and c = 8.
02

Set up the factored form

The factored form of a quadratic polynomial is \[ (mx + n)(px + q) \]. Based on the coefficient a=3, we can assume m and p to be factors of 3, which are 3 and 1. So, we guess the factors as \[ (3x + n)(x + q) \].
03

Find pairs of factors for ac

The product of a and c is 3 * 8 = 24. Find pairs of factors of 24 that add up to the middle coefficient b, which is -25. The pairs are: (-1, -24), (1, 24), (2, 12), (3, 8), (-3, -8), etc.
04

Select the appropriate factor pairs

Among the factor pairs, we find that -1 and -24 add up to -25: \[-1 + (-24) = -25\]
05

Substitute and solve

Write the polynomial as \[ 3x^2 - x - 24x + 8\] and group terms to factor by grouping: \[ x(3x - 1) - 8(3x - 1) = (3x - 1)(x - 8)\]

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Key Concepts

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

quadratic polynomial
A quadratic polynomial is a type of polynomial that features a term raised to the second power. Generally, it can be written in the form: \[ ax^2 + bx + c \] where `a`, `b`, and `c` are coefficients and `a` is not equal to zero. In the context of our example, the quadratic polynomial is \[ 3x^2 - 25x + 8 \] This equation represents a parabola when graphed. The quadratic term (\(3x^2\)) is what gives the parabola its characteristic 鈥楿鈥� shape.
  • The coefficient `a` (3) affects the width and direction of the parabola.
  • The coefficient `b` (-25) affects the location of the vertex and the axis of symmetry.
  • The constant `c` (8) determines the y-intercept of the graph.
Understanding the role of each coefficient can help you better grasp how quadratic polynomials behave under different transformations.
trial-and-error method
The trial-and-error method is a systematic way to factor quadratic polynomials. Instead of relying solely on formulas, this method involves testing potential factor pairs until the correct ones are found. To factor \[ 3x^2 - 25x + 8 \] we follow these steps:
  • Identify the coefficients: `a` (3), `b` (-25), and `c` (8).
  • Set up the factored form: \[ (mx + n)(px + q) \] Since `a` is 3, we use factor pairs of 3: (3, 1).
  • Find pairs of factors for the product of `a` and `c` (3 * 8 = 24).
  • Check factor pairs that add up to -25: (-1, -24).
  • Rewrite and factor by grouping: \[ 3x^2 - x - 24x + 8 \rightarrow x(3x - 1) - 8(3x - 1) \rightarrow (3x - 1)(x - 8) \]
This method may seem tedious initially, but it builds understanding and skills in factoring polynomials manually.
factor pairs
In the process of factoring quadratic polynomials, identifying factor pairs is crucial. Factor pairs are sets of two numbers that, when multiplied together, give the product of `a` and `c` in the quadratic polynomial. For example, in \[ 3x^2 - 25x + 8 \] the product of coefficients `a` and `c` is 24 (3 * 8). The possible factor pairs of 24 include:
  • (1, 24)
  • (2, 12)
  • (3, 8)
  • (4, 6)
  • (-1, -24)
  • (-2, -12)
  • (-3, -8)
  • (-4, -6)
Among these factor pairs, we look for those that add up to the middle coefficient `b` (-25). The pair (-1, -24) works because \[ -1 + (-24) = -25 \] Identifying the correct factor pairs allows us to rewrite the polynomial and factor by grouping, ensuring effective factorization.

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