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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2:04 minutes

Problem 137

Textbook Question

Find all values of b or c that will make the polynomial a perfect square trinomial. 4z^2+bz+81

Verified step by step guidance

A perfect square trinomial takes the form \((az + c)^2 = a^2z^2 + 2acz + c^2\).

In the given polynomial \(4z^2 + bz + 81\), identify \(a^2\) and \(c^2\) from the terms \(4z^2\) and \(81\).

Notice that \(4z^2\) is \((2z)^2\), so \(a = 2\), and \(81\) is \(9^2\), so \(c = 9\).

The middle term of a perfect square trinomial is \(2acz\). Substitute \(a = 2\) and \(c = 9\) to find \(b = 2 \times 2 \times 9\).

Calculate \(b\) to verify the middle term \(bz\) matches \(2acz\) for the polynomial to be 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 expressed as the square of a binomial. It takes the form (ax + b)² = a²x² + 2abx + b². For a trinomial to be a perfect square, the middle term must equal twice the product of the square roots of the first and last terms.

Quadratic Coefficients

In a quadratic expression of the form ax² + bx + c, the coefficients a, b, and c play crucial roles in determining the shape and properties of the parabola represented by the equation. Specifically, the coefficient b is related to the linear term, which influences the vertex and axis of symmetry of the parabola.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This involves manipulating the expression to create a binomial square, which can simplify solving equations or analyzing the function. It is particularly useful for finding the vertex of a parabola or solving quadratic equations.