Textbook Question
In Exercises 133–136, factor each polynomial completely. Assume that any variable exponents represent whole numbers.36x²ⁿ − y²ⁿ
785
views
0. Review of Algebra
Factoring Polynomials
2:38 minutesProblem 139
Textbook Question
Find all values of b or c that will make the polynomial a perfect square trinomial. 100r2-60r+c
Verified step by step guidance1
Recognize that a perfect square trinomial takes the form \(\left(\sqrt{A}r + d\right)^2 = Ar^2 + 2d\sqrt{A}r + d^2\), where \(A\) is the coefficient of \(r^2\) and \(d\) is a constant to be determined.
Identify the coefficient of \(r^2\) in the given polynomial: \$100r^2\(. So, \)A = 100$ and \(\sqrt{A} = 10\).
Compare the middle term of the polynomial, \(-60r\), to the middle term of the perfect square form, \(2d \times 10r = 20d r\). Set \$20d = -60\( to solve for \)d$.
Solve for \(d\) by dividing both sides of the equation \$20d = -60\( by 20, which gives \)d = -3$.
Find the constant term \(c\) by squaring \(d\): \(c = d^2 = (-3)^2\). This value of \(c\) will make the polynomial a perfect square trinomial.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey 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 (ax + b)^2 = a^2x^2 + 2abx + b^2. Recognizing this form helps identify the necessary constant term to complete the square.
Recommended video:
06:24
Solving Quadratic Equations by Completing the Square
Identifying Coefficients in Quadratics
In a quadratic expression ax^2 + bx + c, the coefficients a, b, and c determine its shape and factorization. Understanding how these coefficients relate, especially the middle term and constant, is essential to determine if the polynomial is a perfect square.
Recommended video:
05:35Introduction to Quadratic Equations
Completing the Square
Completing the square involves rewriting a quadratic expression so that it forms a perfect square trinomial. This process often requires finding a specific constant term c that makes the expression factorable as (mx + n)^2, which is key to solving the given problem.
Recommended video:
06:24Solving Quadratic Equations by Completing the Square
Related Videos
Related Practice