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Chapter 6: Problem 27

Factor completely. If the polynomial cannot be factored, write prime. $$ b^{2}+8 b+15 $$

Short Answer

Expert verified
(b + 3)(b + 5)

Step by step solution

01

- Identify the Polynomial Form

The given polynomial is in the form of a quadratic expression: \( b^{2} + 8b + 15 \).
02

- Apply the Factoring Formula

To factor the quadratic expression, look for two numbers that multiply to the constant term (15) and add up to the coefficient of the linear term (8).
03

- Find the Pair of Numbers

Identify the pair of numbers that multiply to 15 and add to 8. These numbers are 3 and 5 because: \( 3 \times 5 = 15 \) \( 3 + 5 = 8 \).
04

- Write the Factored Form

Rewrite the polynomial in its factored form using the identified pair of numbers: \( b^{2} + 8b + 15 = (b + 3)(b + 5) \). Confirm the factorization by expanding the expression to ensure it matches the original polynomial.

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Key Concepts

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

Quadratic Expressions
A quadratic expression is a polynomial of degree 2. This means the highest exponent of the variable (like 'b') is 2.

Quadratic expressions take the form: \[ ax^2 + bx + c \].

In our example, the quadratic expression is \[ b^2 + 8b + 15 \].

Quadratic expressions are useful in many areas like physics, engineering, and economics. Identifying them correctly is the first step in solving them.
Factoring Polynomials
Factoring is breaking down a polynomial into simpler polynomials that, when multiplied together, give back the original polynomial.

In this problem, you need to factor \[ b^2 + 8b + 15 \].

Factoring polynomials helps make complex problems simpler. For instance, it transforms a challenging quadratic expression into a product of linear factors.
Factoring Techniques
There are different ways to factor polynomials. Here, we use the 'factor by grouping' method:
  • Identify a pair of numbers whose product is the constant term (here, 15) and whose sum is the coefficient of the linear term (here, 8).
  • In this example, those numbers are 3 and 5 because 3 times 5 is 15 and 3 + 5 equals 8.
  • Rewrite the polynomial as a product of linear factors: \[ (b + 3)(b + 5) \].

Confirm the factorization by expanding it back to the original polynomial. If you did it correctly, the expanded form should match the original quadratic expression \[ b^2 + 8b + 15 \].
Algebraic Expressions
An algebraic expression is a combination of variables, numbers, and operators (like +, -, *, and /). Quadratic expressions are a specific type of algebraic expression.

Understanding how to manipulate algebraic expressions is essential for problem-solving in math. This includes simplifying, factoring, and expanding expressions.

In this exercise, by factoring the quadratic expression \[ b^2 + 8b + 15 \], you practice these skills. The final factored form, \[ (b + 3)(b + 5) \], shows we've turned a complex expression into simpler parts.

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Most popular questions from this chapter

Factor completely. If the polynomial cannot be factored, write prime. $$ x^{2}+3 x-39 $$

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Solve each equation. $$ 2 t+10=0 $$

Factor completely. $$ x^{2}+4 x y+3 y^{2} $$

Factor each trinomial completely. $$ 3 y^{2}+48 y+192 $$
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