Textbook Question
Use a number line to determine whether each statement is true or false. -4 > -3
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Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem 67
Factor each polynomial completely. See Example 6. 8x³y⁴ + 12x²y³ + 36xy⁴
Verified step by step guidance1
Identify the greatest common factor (GCF) of all the terms in the polynomial. Look at the coefficients (8, 12, 36), the powers of x (x³, x², x¹), and the powers of y (y⁴, y³, y⁴).
Factor out the GCF from each term. For the coefficients, find the largest number that divides 8, 12, and 36. For the variables, take the lowest power of x and y that appears in all terms.
Rewrite the polynomial as the product of the GCF and the remaining polynomial inside parentheses. This means expressing the original polynomial as \(\text{GCF} \times (\text{remaining terms})\).
Simplify the terms inside the parentheses by dividing each original term by the GCF you factored out.
Check if the polynomial inside the parentheses can be factored further. If it can, continue factoring; if not, the factorization is complete.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Greatest Common Factor (GCF)
The Greatest Common Factor is the largest expression that divides all terms of a polynomial without leaving a remainder. Identifying the GCF helps simplify polynomials by factoring it out, making the remaining expression easier to work with.
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Factoring
Factoring Polynomials
Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or monomials. This process often starts by extracting the GCF, followed by factoring special products or applying methods like grouping to break down the expression completely.
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Exponents and Variables in Factoring
When factoring terms with variables and exponents, it is important to consider the lowest power of each variable common to all terms. Factoring out variables involves subtracting exponents according to the laws of exponents, ensuring the factored form is accurate and simplified.
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Related Practice
Textbook Question
Simplify each radical. See Example 5. √24
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Textbook Question
Use the product and quotient rules for radicals to rewrite each expression. See Example 4. 30√10 / 5√2
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Textbook Question
Add or subtract, as indicated. See Example 4. (3x)/(x² + x − 12) − x/(x² − 16) + x
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Textbook Question
Use a number line to determine whether each statement is true or false. See Example 6. -6 < -1
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Textbook Question
Use the graph of y = ƒ(x) to find each function value: (a) ƒ(-2) (b) ƒ(0) (c) ƒ(1) and (d) ƒ(4). See Example 7(d).
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