Textbook Question
Simplify the algebraic expression. 5(2x - 3) + 7x
Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 1, Problem 18
In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x2+8x+15
Verified step by step guidance1
Identify the trinomial in the form \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = 8 \), and \( c = 15 \).
Look for two numbers that multiply to \( c = 15 \) and add to \( b = 8 \). These two numbers will be used to split the middle term.
The two numbers that satisfy the conditions are 5 and 3, because \( 5 \times 3 = 15 \) and \( 5 + 3 = 8 \).
Rewrite the trinomial by splitting the middle term using the two numbers found: \( x^2 + 5x + 3x + 15 \).
Group the terms into two pairs and factor each group: \( (x^2 + 5x) + (3x + 15) \), then factor out the greatest common factor (GCF) from each group.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Trinomials
Factoring trinomials involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. This process requires identifying two numbers that multiply to 'c' (the constant term) and add to 'b' (the coefficient of the linear term). For example, in the trinomial x^2 + 8x + 15, we look for two numbers that multiply to 15 and add to 8.
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Prime Trinomials
A trinomial is considered prime if it cannot be factored into the product of two binomials with real coefficients. This typically occurs when the discriminant of the quadratic equation is negative or when no integer pairs satisfy the conditions for factoring. Recognizing prime trinomials is essential for determining whether a quadratic expression can be simplified further.
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Quadratic Formula
The quadratic formula, given by x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of a quadratic equation ax^2 + bx + c = 0. While not directly used for factoring, it helps determine if a trinomial can be factored by revealing the nature of its roots. If the roots are rational, the trinomial can be factored; if they are irrational or complex, it may be prime.
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Related Practice
Textbook Question
Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers.
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Textbook Question
State the name of the property illustrated: (3 • 7) + (4 • 7) = (4 • 7) + (3 •7)
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Textbook Question
In Exercises 15–32, multiply or divide as indicated. (x2−4)/(x2−4x+4) ⋅ (2x−4)/(x+2)
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Textbook Question
A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula h=4+60t-16t2 describes the ball's height above the ground, h, in feet, t seconds after it was kicked. Use this formula to solve Exercises 19–20. What was the ball's height 2 seconds after it was kicked?
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Textbook Question
Find each product. (x+7)(x+3)
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