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Ch. R - Algebra Review
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. R - Algebra ReviewProblema 93
Capitolo 1, Problema 93

Factor each polynomial completely. See Example 6. t⁴ - 1

Guida verificata passo dopo passo
1
Recognize that the polynomial \(t^{4} - 1\) is a difference of squares because it can be written as \((t^{2})^{2} - 1^{2}\).
Apply the difference of squares formula: \(a^{2} - b^{2} = (a - b)(a + b)\), so rewrite \(t^{4} - 1\) as \((t^{2} - 1)(t^{2} + 1)\).
Notice that \(t^{2} - 1\) is itself a difference of squares and can be factored further as \((t - 1)(t + 1)\).
The term \(t^{2} + 1\) is a sum of squares, which does not factor over the real numbers, so leave it as is.
Combine all factors to express the complete factorization: \((t - 1)(t + 1)(t^{2} + 1)\).

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Difference of Squares

The difference of squares is a factoring technique used when an expression is in the form a² - b². It factors into (a - b)(a + b). For example, t⁴ - 1 can be seen as (t²)² - 1², allowing it to be factored using this method.
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Sum and Difference of Tangent

Factoring Higher Powers

Polynomials with higher powers, like t⁴, can often be factored by recognizing patterns such as squares of squares or sums/differences of powers. Breaking down t⁴ - 1 into (t² - 1)(t² + 1) simplifies the expression further.
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Factoring

Factoring Quadratic Expressions

After applying difference of squares, resulting quadratic expressions like t² - 1 can be factored further if they are also differences of squares. This stepwise factoring continues until the polynomial is completely factored into irreducible factors.
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Factoring
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Add or subtract, as indicated. See Example 6. 5√3 + √12

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Factor each polynomial completely. See Example 6. 8t³ + 125

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Simplify each inequality if needed. Then determine whether the statement is true or false. -6 < 7 + 3

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Simplify each complex fraction. See Examples 5 and 6. [(y + 3)/y − 4/(y − 1)] / [(y/y + 1/y) / (y − 1) + 1/y]

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(Modeling) Distance from the Origin of the Nile River The Nile River in Africa is about 4000 mi long. The Nile begins as an outlet of Lake Victoria at an altitude of 7000 ft above sea level and empties into the Mediterranean Sea at sea level (0 ft). The distance from its origin in thousands of miles is related to its height above sea level in thousands of feet (x) by the following formula.

Distance = (7 − x) / (0.639x + 1.75)

For example, when the river is at an altitude of 600 ft, x = 0.6 (thousand), and the distance from the origin is

Distance ≈ (7 − 0.6) / (0.639 × 0.6 + 1.75) ≈ 3, which represents 3000 mi. (Data from The World Almanac and Book of Facts.) What is the distance from the origin of the Nile when the river has an altitude of 7000 ft?

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Add or subtract, as indicated. See Example 6. √6 + √7

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