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Chapter 6: Problem 42

Write in factored form by factoring out the greatest common factor. \(21 b^{3}+7 b^{2}\)

Short Answer

Expert verified
7b^2 (3b + 1)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Determine the greatest common factor of the terms in the expression. For the terms 21 and 7, the GCF is 7. The variable part of both terms is the lowest power of b, which is b^2. Hence, the GCF of the entire expression is 7b^2.
02

Factor Out the GCF

Rewrite the expression by factoring out the GCF (7b^2). This results in: \[ 21b^3 + 7b^2 = 7b^2 (3b) + 7b^2 (1) \]
03

Simplify

Combine the terms inside the parentheses:\[ 7b^2 (3b + 1) \]

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Key Concepts

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

greatest common factor
The first step in polynomial factoring is identifying the Greatest Common Factor (GCF). This is the largest factor that can divide all terms in the polynomial without leaving a remainder.

For numerical coefficients, find the GCF by comparing the numbers' prime factors or simply by recognizing common divisors.

For example, in the polynomial expression `21b^3 + 7b^2`, we need to find the GCF of 21 and 7. It's easy to see that the GCF is 7.

For variable parts, take the lowest power of the variable present in all terms. Here, we have `b^3` and `b^2`, so the GCF is `b^2`.

Combining these, the GCF for the entire expression is `7b^2`.

Key points:
  • Identify the largest number that divides all coefficients.
  • Use the lowest exponent of the common variable(s).
polynomial factoring
Factoring polynomials involves breaking down the polynomial into simpler terms that, when multiplied together, give back the original polynomial. This is essential in simplifying expressions and solving equations.

Once we identify the GCF, we factor it out from the polynomial.

For the example polynomial `21b^3 + 7b^2`, we found that the GCF is `7b^2`.

We then rewrite the polynomial as `7b^2` multiplied by the remaining terms.

This gives us:

\[ 21b^3 + 7b^2 = 7b^2 (3b) + 7b^2 (1) \]

Finally, we combine the terms inside the parentheses:

\[ 7b^2 (3b + 1) \]

Thus, the polynomial is factored as `7b^2 (3b + 1)`.

Benefits of Polynomial Factoring:
  • Simplifies solving equations.
  • Reduces complex expressions.
  • Reveals roots of polynomial equations.
algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It forms the foundation for higher-level math and various applications in science and engineering.

Understanding algebraic concepts is crucial for solving equations, simplifying expressions, and understanding the relationships between variables.

In our example, we applied key algebraic principles to factor the polynomial:
  • Identifying the Greatest Common Factor (GCF).
  • Rewriting the expression using the GCF.
  • Simplifying the expression.


Let's summarize these steps in the context of our polynomial `21b^3 + 7b^2`:

  • Identify the GCF: `7b^2`.
  • Factor out the GCF: \[ 21b^3 + 7b^2 = 7b^2 (3b) + 7b^2 (1) \].
  • Simplify the expression: \[ 7b^2 (3b + 1) \].


By mastering these steps and fundamentals of algebra, you can tackle a wide range of mathematical problems with confidence!

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Most popular questions from this chapter

Apply the special factoring nules of this section to factor each binomial or trinomial. $$ p^{2}-\frac{1}{9} $$

Solve each equation, and check your solutions. $$ 6 x^{2}(2 x+3)=4(2 x+3)+5 x(2 x+3) $$

Find each product. $$ (3 a+2)(2 a+1) $$

Factor completely. $$ 2 x^{6}+8 x^{5}-42 x^{4} $$

Brain Busters Factor each polynomial. ( Hint: As the first step, factor out the greatest common factor.) $$ 25 q^{2}(m+1)^{3}-5 q(m+1)^{3}-2(m+1)^{3} $$
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