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Chapter 5: Problem 147

Factor each polynomial completely. $$a^{2}+10 a+25$$

Short Answer

Expert verified
(a + 5)^{2}

Step by step solution

01

Identify the polynomial

First, recognize the polynomial given: a^{2} + 10a + 25This is a quadratic polynomial of the form ax^{2} + bx + c
02

Recognize the perfect square trinomial

Notice that the polynomial can be written as a perfect square trinomial. We look for numbers that satisfy the form (a+b)^{2} = a^{2} + 2ab + b^{2}In this exercise, compare the given polynomial with the form above.
03

Find a and b

To accomplish this, identify the values:a^{2} is a^{2} 2ab is 10ab^{2} is 25Hence, a = a and b = 5
04

Write the polynomial as a square of a binomial

Since (a + 5)^{2} = a^{2} + 2(5)a + 5^{2} = a^{2} + 10a + 25The factored form is (a + 5)^{2}

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Key Concepts

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

Perfect Square Trinomial
A perfect square trinomial is a special form of a quadratic polynomial which can be factored into the square of a binomial.

The general form of a perfect square trinomial is \((a + b)^{2} = a^{2} + 2ab + b^{2}\). Notice how the middle term, \(2ab\), is twice the product of the two terms inside the binomial.

In the given polynomial \(a^{2} + 10a + 25\), it can be rewritten as \((a + 5)^{2}\) because it fits the form \(a^{2} + 2ab + b^{2}\). Here,\(a\) is the same as in the binomial, and \(b\) is 5.

To factor any such polynomial, follow these steps:
  • Identify the form \(a^{2} + 2ab + b^{2}\).
  • Find the values of \(a\) and \(b\) such that when squared and doubled they match the original terms.
  • Rewrite the polynomial as \((a + b)^{2}\).
Factoring perfect square trinomials can simplify solving quadratic equations and make it easier to graph polynomials.
Quadratic Polynomial
A quadratic polynomial is a polynomial of degree 2. It has the general form \(ax^{2} + bx + c\). In our example, the polynomial \(a^{2} + 10a + 25\) is a quadratic polynomial where \(a = 1\), \(b = 10\), and \(c = 25\).

Quadratic polynomials have important properties that make them easier to work with:
  • They always graph to a parabola.
  • The sign and magnitude of \(a\) determine the direction and width of the parabola.
There are various methods to factor quadratic polynomials. For example:
  • Factoring by grouping.
  • Applying the quadratic formula.
  • Recognizing perfect square trinomials.
In this exercise, we used the recognition method for a perfect square trinomial, which is a quicker way to factor specific types of quadratic polynomials.
Binomial
A binomial is a polynomial with exactly two terms. Examples include expressions like \(a + b\) or \(x - 3\). When we square a binomial, we get a quadratic polynomial.

In other words, \((a + b)^{2} = a^{2} + 2ab + b^{2}\). This transformation from a binomial to a perfect square trinomial is often utilized in polynomial factorization.

For our problem, the given polynomial \(a^{2} + 10a + 25\) factors into \((a + 5)^{2}\). Here, \(a\) is the variable and 5 is the constant term.

Breaking it down:
  • First, we find the possible binomial form, checking if the polynomial fits \(a^{2} + 2ab + b^{2}\).
  • Then, we identify the terms \(a\) and \(b\).
  • Lastly, we rewrite the expression as the square of the binomial.
Recognizing these patterns is fundamental in algebra, as it aids in simplifying expressions and solving equations more efficiently.

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