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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 137
Chapter 1, Problem 137

Find all values of b or c that will make the polynomial a perfect square trinomial. 4z2+bz+81

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1
Recall that a perfect square trinomial can be written in the form \(\left( mz + n \right)^2 = m^2 z^2 + 2mnz + n^2\).
Compare the given polynomial \(4z^2 + bz + 81\) to the general form \(m^2 z^2 + 2mnz + n^2\) to identify \(m^2\) and \(n^2\).
Since the coefficient of \(z^2\) is 4, set \(m^2 = 4\), which gives \(m = 2\) (considering the positive root for simplicity).
Since the constant term is 81, set \(n^2 = 81\), which gives \(n = 9\) (again, considering the positive root).
Use the middle term formula \$2mnz\( to find \)b\(: \(b = 2 \times m \times n = 2 \times 2 \times 9\), and write the value of \)b$ that makes the polynomial a perfect square trinomial.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, typically in the form (x + d)^2 = x^2 + 2dx + d^2. Recognizing this form helps identify conditions on coefficients to make the polynomial a perfect square.
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Comparing Coefficients

To determine values of variables that make a polynomial a perfect square, compare the given polynomial's coefficients with those of the expanded perfect square form. This method allows solving for unknown coefficients by matching terms.
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Example 3

Factoring Quadratic Expressions

Factoring quadratics involves rewriting the expression as a product of binomials. Understanding how to factor and expand quadratics is essential to verify if a polynomial is a perfect square and to find the necessary coefficient values.
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