Table of contents

0. Review of Algebra4h 18m

Algebraic Expressions
36m

Exponents
32m

Polynomials Intro
19m

Multiplying Polynomials
36m

Factoring Polynomials
1h 2m

Radical Expressions
15m

Simplifying Radical Expressions
35m

Rationalize Denominator
15m

Rational Exponents
4m

1. Equations & Inequalities3h 18m

Linear Equations
31m

Rational Equations
21m

The Imaginary Unit
6m

Powers of i
11m

Complex Numbers
36m

Intro to Quadratic Equations
18m

The Square Root Property
11m

Completing the Square
12m

The Quadratic Formula
18m

Choosing a Method to Solve Quadratics
9m

Linear Inequalities
20m

2. Graphs of Equations1h 43m

Graphs and Coordinates
7m

Two-Variable Equations
23m

Lines
1h 12m

3. Functions2h 17m

Intro to Functions & Their Graphs
35m

Common Functions
5m

Transformations
45m

Function Operations
21m

Function Composition
29m

4. Polynomial Functions1h 44m

Quadratic Functions
39m

Understanding Polynomial Functions
34m

Graphing Polynomial Functions
30m

Dividing Polynomials

Zeros of Polynomial Functions

5. Rational Functions1h 23m

Introduction to Rational Functions
9m

Asymptotes
35m

Graphing Rational Functions
38m

6. Exponential & Logarithmic Functions2h 28m

Introduction to Exponential Functions
9m

Graphing Exponential Functions
25m

The Number e
8m

Introduction to Logarithms
22m

Graphing Logarithmic Functions
18m

Properties of Logarithms
27m

Solving Exponential and Logarithmic Equations
35m

7. Systems of Equations & Matrices4h 5m

Two Variable Systems of Linear Equations
1h 7m

Introduction to Matrices
1h 11m

Determinants and Cramer's Rule
1h 3m

Graphing Systems of Inequalities
42m

8. Conic Sections2h 23m

Introduction to Conic Sections
5m

Circles
15m

Ellipses: Standard Form
33m

Parabolas
36m

Hyperbolas at the Origin
40m

Hyperbolas NOT at the Origin
12m

9. Sequences, Series, & Induction1h 22m

Sequences
40m

Arithmetic Sequences
25m

Geometric Sequences
15m

10. Combinatorics & Probability1h 45m

Factorials
11m

Combinatorics
46m

Probability
47m

0. Review of Algebra

Factoring Polynomials

College Algebra

0. Review of Algebra

Factoring Polynomials

Previous problemNext problem

2:04 minutes

Problem 137

Textbook Question

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

Verified step by step guidance

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.

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.

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.