Ch. R - Algebra Review

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. R - Algebra Review

Problema R.3.7

Capitolo 1, Problema R.3.7

CONCEPT PREVIEW Work each problem. Match each polynomial in Column I with its factored form in Column II. I II a. 8x³ - 27 A. (3 - 2x) (9 + 6x + 4x²) b. 8x³ + 27 B. (2x - 3) (4x² + 6x + 9) c. 27 - 8x³ C. (2x + 3) (4x² - 6x + 9)

Guida verificata passo dopo passo

Recognize that each polynomial in Column I is a difference or sum of cubes. Recall the formulas for factoring cubes: for difference of cubes, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, and for sum of cubes, $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Identify $a$ and $b$ for each polynomial by expressing the terms as cubes. For example, \$8x^3$ can be written as $(2x)^3$ and $27$ as \$3^3$.

Apply the difference of cubes formula to $8x^3 - 27$ and $27 - 8x^3$, noting the order of subtraction affects the signs in the factors.

Apply the sum of cubes formula to $8x^3 + 27$, carefully substituting $a = 2x$ and $b = 3$ into the formula.

Match each factored form from Column II to the corresponding polynomial in Column I by comparing the factors obtained from the formulas with the given options.

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Sum and Difference of Cubes

The sum and difference of cubes are special polynomial forms that factor into binomial and trinomial products. For example, a³ + b³ factors as (a + b)(a² - ab + b²), and a³ - b³ factors as (a - b)(a² + ab + b²). Recognizing these forms helps in quickly factoring cubic expressions.

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Sum and Difference of Tangent

Identifying 'a' and 'b' in Cubic Expressions

To factor cubic polynomials using sum or difference of cubes, it is essential to identify the cube roots of each term. For instance, in 8x³, the cube root is 2x, and in 27, it is 3. Correctly determining these values allows proper application of the factoring formulas.

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Simplifying Trig Expressions

Matching Factored Forms to Original Polynomials

After factoring, it is important to verify the factored form by expanding it back to the original polynomial. This ensures the correct pairing between the polynomial and its factorization, especially when multiple similar options are given, as in matching problems.

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Position Vectors & Component Form

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CONCEPT PREVIEW Work each problem. Match each polynomial in Column I with its factored form in Column II. I II a. x² + 10xy + 25y² A. (x + 5y) (x - 5y) b. x² - 10xy + 25y² B. (x + 5y)² c. x² - 25y² C. (x - 5y)² d. 25y² - x² D. (5y + x) (5y - x)

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Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. -(4m³n⁰)²

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Domanda del libro di testo

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. - ( x³y⁵/z)⁰

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Domanda del libro di testo

CONCEPT PREVIEW Fill in the blank to correctly complete each sentence. The polynomial 2x⁵ - x + 4 is a trinomial of degree _________.

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Domanda del libro di testo

Match each expression in Column I with its equivalent in Column II. See Example 3. I II. a. 6° A. 0 b. -6° B. 1 c. (-6)° C. -1 d. -(-6)° D. 6 E. -6

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Domanda del libro di testo

Simplify each expression. See Example 1. (-4x⁵) (4x²)

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