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Chapter 6: Problem 39

Factor completely. $$ 5 a^{2}-10 a b+5 b^{2} $$

Short Answer

Expert verified
5(a - b)^2

Step by step solution

01

- Identify the common factor

First, check and identify if there is a common factor in each term of the expression. Here, each term in the expression has a common factor of 5.
02

- Factor out the common factor

Factor out the common factor 5 from each term of the expression. This gives: 5(a^2 - 2ab + b^2)
03

- Recognize the quadratic form

Now observe the quadratic expression inside the parentheses: a^2 - 2ab + b^2. This is a perfect square trinomial.
04

- Factor the perfect square trinomial

The expression inside the parentheses can be factored as the square of a binomial. Notice that a^2 - 2ab + b^2 can be written as (a - b)^2.
05

- Write the final factored form

Combine the factored common factor with the factored trinomial. The fully factored form is 5(a - b)^2

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Key Concepts

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

Common Factor
A common factor is a number or algebraic term that divides each term in an expression without leaving a remainder.

Identifying the common factor is often the first step in factoring any polynomial.
In our exercise, we recognized that the common factor among the terms in the expression \( 5a^2 - 10ab + 5b^2 \) is 5.

After identifying the common factor, we factor it out to simplify the expression further.
This gives us: \( 5(a^2 - 2ab + b^2) \).
Perfect Square Trinomial
A perfect square trinomial is a quadratic expression that can be written as the square of a binomial.

Perfect square trinomials take the form: \( a^2 \text{+ or -} 2ab + b^2 \).
In our example, the expression inside the parentheses, \( a^2 - 2ab + b^2 \), follows this pattern precisely.

To recognize a perfect square trinomial:
  • Square the first term to get \( a^2 \)
  • Square the last term to get \( b^2 \)
  • Confirm that the middle term is twice the product of the two terms, hence \( -2ab \).
In this case, the quadratic form inside the parentheses is indeed a perfect square trinomial.
Quadratic Expression
A quadratic expression is a polynomial of degree 2 and typically takes the general form \( ax^2 + bx + c \).

Quadratic expressions are central to algebra and factoring them is a crucial skill.
Our example, \( 5(a^2 - 2ab + b^2) \), contains the quadratic expression \( a^2 - 2ab + b^2 \).

When working with quadratics, always consider the possibility they might be a perfect square trinomial.
This helps in efficiently simplifying complicated expressions.
Binomial
A binomial is a polynomial with exactly two terms.

In the context of factoring polynomials, recognizing binomials within expressions is pivotal.
In our example, the quadratic expression \( a^2 - 2ab + b^2 \) was identified as a perfect square trinomial that could be factored as the square of a binomial: \( (a - b)^2 \).

To factor it correctly:
  • Identify the first term: \( a \)
  • Identify the second term: \( b \)
  • Observe that the middle term fits the form \( -2ab \)
  • Combine them into a binomial squared: \( (a - b)^2 \)
Finally, our fully factored expression is \( 5(a - b)^2 \).
This process illustrates how understanding binomials helps simplify and solve quadratic polynomials.

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Most popular questions from this chapter

Write an equivalent expression by factoring. Assume that all exponents are natural numbers. $$ 3 a^{n+1}+6 a^{n}-15 a^{n+2} $$

Factor completely. Assume that variables in exponents represent positive integers. $$ s^{2}-4 s t+4 t^{2}+4 s-8 t+4 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 16-a^{2}-2 a b-b^{2} $$

Review multiplying binomials using FOIL $$ (x+2)(x+7) $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ \frac{1}{49}-x^{2} $$
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