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Chapter 6: Problem 38

Factor completely. If the polynomial cannot be factored, write prime. $$ q^{2}-q-42 $$

Short Answer

Expert verified
\( q^2 - q - 42 = (q - 7)(q + 6) \)

Step by step solution

01

Identify the quadratic

Observe that the given polynomial is in standard quadratic form: \[ q^2 - q - 42 \]
02

Set up the factoring form

A quadratic in the form \[ ax^2 + bx + c \] can often be factored as \[ (x - p)(x - q) \]. Here, we need to find two numbers that multiply to \[ c = -42 \] and add to \[ b = -1 \].
03

Find the factors

We need to find two numbers that multiply to -42 and add to -1. Considering the pairs of factors of -42, we see that the pair \[ -7 \times 6 = -42 \] and \[ -7 + 6 = -1 \]. So, the factors are -7 and 6.
04

Write the factorization

Based on the factors found in Step 3, rewrite the polynomial as:\[ (q - 7)(q + 6) \]

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Key Concepts

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

Quadratic Equations
Quadratic equations are a type of polynomial equation in the form of



ax^2 + bx + c = 0
where:
  • a is the coefficient of the quadratic term ( ax^2 ),

  • b is the coefficient of the linear term ( bx ),

  • and c is the constant term.

    Quadratic equations are useful for describing many natural phenomena.
    We can solve them using various methods, such as the quadratic formula, completing the square, or factoring,
as in the provided exercise. Understanding how to factor quadratic equations can help simplify the solving process and reveal important characteristics about the polynomial.
Factoring
Factoring is a method used in algebra to break down expressions into simpler multiples or factors that, when multiplied together, give the original expression.

In the provided exercise, we factor the quadratic polynomial: q^2 - q - 42, by searching for two numbers that multiply to the constant term (-42) and sum to the coefficient of the linear term (-1).
Here are key points to remember:
  • Identify the quadratic expression in the standard form ax^2 + bx + c.

  • Find factors of the constant term ( c ) that also sum up to the linear coefficient ( b ).

  • Write the polynomial in its factored form using those two numbers.
Factoring is an essential tool in simplifying algebraic expressions and solving polynomial equations.
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its factors. For the quadratic polynomial q^2 - q - 42,
we identified factors that worked as -7 and 6.

Here's how we proceeded:
  • Determine the product needed ( -42 ).

  • Determine the sum needed ( -1 ).

  • Identify the two numbers that fit those requirements ( -7 and 6 ).

  • Rewrite the polynomial in its factorized form ( (q - 7)(q + 6) ).

Factoring polynomials helps in many areas of algebra, such as simplifying expressions and solving higher-degree equations.
Algebraic Expressions
Algebraic expressions include numbers, variables, and operators that represent values and relationships. The quadratic polynomial q^2 - q - 42 is an example of such an expression.
To work with algebraic expressions effectively:
  • Recognize different types of polynomials and their degrees.

  • Understand the importance of standard form ( ax^2 + bx + c ).

  • Apply factoring techniques to simplify the expression or solve related equations.

Familiarity with these concepts allows you to tackle various algebraic problems more efficiently and with greater confidence.

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Most popular questions from this chapter

Factor by grouping. \(3 a^{3}+3 a b^{2}+2 a^{2} b+2 b^{3}\)

Factor each polynomial. $$ (3 m-n) k^{2}-13(3 m-n) k+40(3 m-n) $$

Factor each polynomial. $$ a^{5}+3 a^{4} b-4 a^{3} b^{2} $$

Factor each trinomial completely. $$ 25 z^{4}+5 z^{3}+z^{2} $$

Factor each trinomial completely. $$ 4 k^{3}-4 k^{2}+9 k $$
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