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Chapter 5: Problem 148

Factor each polynomial completely. $$3 a^{2}-22 a+7$$

Short Answer

Expert verified
\((3a - 1)(a - 7)\)

Step by step solution

01

Identify coefficients

Identify the coefficients from the given quadratic polynomial. For the polynomial \(3a^2 - 22a + 7\), the coefficients are: \(a = 3\), \(b = -22\), and \(c = 7\).
02

Multiply coefficient a and c

Multiply the coefficient of \(a^2\) and the constant term (c). That gives: \(a \times c = 3 \times 7 = 21\).
03

Find two numbers that multiply to ac and add to b

Find two numbers that multiply to 21 and add or subtract to give the middle coefficient (-22). The numbers are -21 and -1, because \(-21 \times -1 = 21\) and \(-21 + (-1) = -22\).
04

Rewrite middle term

Rewrite the middle term using the two numbers found: \(3a^2 - 21a - 1a + 7\).
05

Factor by grouping

Group the terms into two pairs and factor out the common factors: \((3a^2 - 21a) - (1a - 7) = 3a(a - 7) - 1(a - 7)\).
06

Factor out the common binomial factor

Factor out the common binomial factor \((a - 7)\): \((a - 7)(3a - 1)\).

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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 polynomial of degree 2, which means that the highest power of the variable is 2. The standard form of a quadratic polynomial is given by: \[ ax^2 + bx + c \] where:
  • a is the coefficient of \( x^2 \)
  • b is the coefficient of \( x \)
  • c is the constant term
In the given exercise, the quadratic polynomial is: \( 3a^2 - 22a + 7 \). Here, the highest power of the variable \( a \) is 2, making it a quadratic polynomial.
coefficients
Coefficients are the numerical factors that multiply the variables in a polynomial. In the quadratic polynomial \( ax^2 + bx + c \), the coefficients are:
  • a (the coefficient of \( x^2 \))
  • b (the coefficient of \( x \))
  • c (the constant term)
For example, in the problem provided:
  • The coefficient of \( a^2 \) is 3.
  • The coefficient of \( a \) is -22.
  • The constant term is 7.
Understanding the coefficients is crucial for solving quadratic polynomials, as they play a significant role in methods like factoring by grouping.
factoring by grouping
Factoring by grouping is a method used to factor polynomials by grouping terms with common factors. Here's how it works: 1. Group terms in pairs:
For the polynomial \( 3a^2 - 22a + 7 \), rewrite the middle term as follows: \( 3a^2 - 21a - a + 7 \). 2. Factor out the greatest common factor (GCF) from each group:
\( 3a(a - 7) - 1(a - 7) \). 3. Factor out the common binomial factor:
\( (a - 7)(3a - 1) \). This technique simplifies the polynomial and makes it easier to factor completely.
binomial factor
A binomial factor is a polynomial with exactly two terms, usually separated by a plus or minus sign. In the factoring process, identifying common binomial factors helps to simplify the polynomial further.For instance, in the provided example:
We identify the common binomial factor \( (a - 7) \) in both groups: \( 3a(a - 7) - 1(a - 7) \).
Factoring out \( (a - 7) \) from both terms results in: \( (a - 7)(3a - 1) \). Recognizing and factoring binomial factors is an essential step in simplifying higher-degree polynomials. It ensures that the polynomial is broken down into its simplest, completely factored form.

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