Skip to main content
Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblema 29
Capitolo 9, Problema 29

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (2a + b)6

Guida verificata passo dopo passo
1
Identify the binomial expression to expand: \((2a + b)^6\).
Recall the Binomial Theorem formula: \(\displaystyle (x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k\), where \(\binom{n}{k}\) is the binomial coefficient.
In this problem, set \(x = 2a\), \(y = b\), and \(n = 6\). The expansion will be \(\sum_{k=0}^6 \binom{6}{k} (2a)^{6-k} b^k\).
Calculate each term by finding the binomial coefficient \(\binom{6}{k}\), raising \$2a$ to the power $(6-k)$, and $b$ to the power $k$, then multiply these values together for each $k$ from 0 to 6.
Simplify each term by applying the exponent to \$2a$ (remember to raise 2 to the power as well), and write the full expanded expression as the sum of all these simplified terms.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
12m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (x + y)^n, where n is a non-negative integer. It states that (x + y)^n equals the sum of terms C(n, k) * x^(n-k) * y^k, where C(n, k) are binomial coefficients. This theorem simplifies the expansion process without multiplying the binomial repeatedly.
Video consigliato:
03:41
Special Products - Cube Formulas

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from a set of n elements. They appear as coefficients in the binomial expansion and can be calculated using factorials or Pascal's Triangle. These coefficients determine the weight of each term in the expanded expression.
Video consigliato:
03:41
Special Products - Cube Formulas

Exponent Rules in Expansion

When expanding (2a + b)^6, each term involves powers of 2a and b whose exponents add up to 6. The exponent rules state that (xy)^n = x^n * y^n and (x^m)^n = x^(m*n). Applying these rules helps simplify each term by correctly raising coefficients and variables to their respective powers.
Video consigliato:
7:39
Introduction to Exponent Rules
Pratica correlata
Domanda del libro di testo

Find each indicated sum. ∑i=165i\(\sum\)_{i=1}^{6} 5i

845
views
Domanda del libro di testo

Use the Fundamental Counting Principle to solve Exercises 29–40. A popular brand of pen is available in three colors (red, green, or blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand?

643
views
Domanda del libro di testo

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for an to find a20, the 20th term of the sequence. a1=-20, d = -4

975
views
Domanda del libro di testo

Use mathematical induction to prove that each statement is true for every positive integer n. ∑i=1n5⋅6i=6(6n−1)\(\sum\)_{i=1}^{n} 5 \(\cdot\) 6^i = 6(6^n - 1)

736
views
Domanda del libro di testo

Write the first three terms in each binomial expansion, expressing the result in simplified form. (x+2)8

689
views
Domanda del libro di testo

Use the Fundamental Counting Principle to solve Exercises 29–40. The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or sport). In how many ways can you order the car?

720
views