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Chapter 5: Problem 147

Factor each polynomial using the trial-and-error method. $$ b^{2}+6 b-16 $$

Short Answer

Expert verified
The factored form is \((b + 8)(b - 2)\).

Step by step solution

01

Identify the Polynomial Structure

The given polynomial is in the form of a quadratic expression: \( b^{2} + 6b - 16 \). Our goal is to factor it into two binomials of the form \((b + m)(b + n)\).
02

Determine the Signs in the Binomials

Since the constant term (-16) is negative and the middle term (6b) is positive, one of the factors in our binomials must be positive and the other must be negative.
03

Find Factor Pairs of the Constant Term

List pairs of factors of -16: (1, -16), (-1, 16), (2, -8), (-2, 8), (4, -4), (-4, 4). Identify which pair of factors add up to the middle coefficient (6).
04

Choose the Correct Pair of Factors

The correct pair is (8, -2) because 8 + (-2) = 6.
05

Write the Factored Form

Using the correct pair of factors, write the binomials: \((b + 8)(b - 2)\). Therefore, \( b^{2} + 6b - 16 = (b + 8)(b - 2) \).

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Key Concepts

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

trial-and-error method
The trial-and-error method is an effective approach when factoring quadratics. It's about trying different factor pairs and checking which one works. This method is especially useful for quadratic expressions of the form \( ax^2 + bx + c \), where \( a = 1 \).
You'll start by listing all the factor pairs of the constant term (in this case, -16).
Here鈥檚 an easy checklist to follow:
  • Write down the quadratic expression.
  • Identify the coefficient pairs of the constant term.
  • Substitute the pairs into the factored form.
  • Check which pair sums up to the middle coefficient.
Keep practicing to get more comfortable with this method, as it can significantly speed up the factoring process!
quadratic expression
A quadratic expression typically looks like \( ax^2 + bx + c \). This means it is a polynomial of degree 2. The quadratic expression in the exercise is \( b^2 + 6b - 16 \).
Here, the coefficient \( a = 1 \), \( b = 6 \), and \( c = -16 \).
To factor a quadratic expression, the goal is to express it as the product of two binomials: \( (b + m)(b + n) \). When expanded, \( (b + m)(b - n) \) should give you back the original quadratic expression.
Understanding this structure helps you know what you are looking for when factoring.
binomials
Binomials are polynomials with two terms. Examples are \( (b + 8) \) and \( (b - 2) \). When you multiply these together, you should get the original quadratic expression.
  • First term (b): This appears in both binomials.
  • Second term (8) and (-2): These add up to the middle term in the quadratic expression.
Remember, factor pairs impact the signs in the binomials. For instance, factors of -16 include both positive and negative numbers, like (8, -2). Practice helps in quickly identifying these pairs and ensuring balance in your biinomials.

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Most popular questions from this chapter

Factor each polynomial by grouping. $$ y^{2}-a y-b y+a b $$

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Factor each polynomial using the trial-and-error method. $$ w^{2}-2 w-15 $$

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