Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Factoring Polynomials

College Algebra

0. Review of Algebra

Factoring Polynomials

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2:38 minutes

Problem 139

Textbook Question

Find all values of b or c that will make the polynomial a perfect square trinomial. 100r^2-60r+c

Verified step by step guidance

Identify the structure of a perfect square trinomial, which is of the form \((ar + b)^2 = a^2r^2 + 2abr + b^2\).

Compare the given polynomial \(100r^2 - 60r + c\) with the perfect square trinomial form \(a^2r^2 + 2abr + b^2\).

Notice that \(a^2 = 100\), so \(a = 10\) (since \(a\) must be positive).

Set \(2ab = -60\) and solve for \(b\) using \(a = 10\), which gives \(2 \times 10 \times b = -60\).

Solve for \(c\) by using \(b^2 = c\) once \(b\) is found.

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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 expressed as the square of a binomial. It takes the form (a ± b)² = a² ± 2ab + b². For a trinomial to be a perfect square, the first term must be a perfect square, the last term must also be a perfect square, and the middle term must be twice the product of the square roots of the first and last terms.

Identifying Coefficients

In the polynomial 100r² - 60r + c, the coefficients are the numerical factors of each term. Here, 100 is the coefficient of r², -60 is the coefficient of r, and c is the constant term. Understanding how to identify and manipulate these coefficients is crucial for determining the conditions under which the polynomial becomes a perfect square trinomial.

Condition for Perfect Square

To determine the value of c that makes the polynomial a perfect square trinomial, we need to ensure that the middle term (-60r) equals twice the product of the square roots of the first term (10r) and the last term (√c). This leads to the equation -60 = 2 * 10 * √c, which can be solved to find the appropriate value of c.