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Chapter 4: Problem 45

In Exercises 43鈥�48, use Pascal鈥檚 Triangle to expand the binomial. \((g+2)^5\)

Short Answer

Expert verified
\(g^5+10g^4+40g^3+80g^2+80g+32\)

Step by step solution

01

Use Pascal's Triangle to Find the Coefficients

The 5th row of Pascal's Triangle (starting from the row \(1\)) is \(1, 5, 10, 10, 5, 1\), and these set of numbers represent the coefficients of the expanded binomial expression.
02

Write down the Binomial Expansion

Following the pattern, one obtains \((g+2)^5 = 1*g^5*2^0+5*g^4*2^1+10*g^3*2^2+10*g^2*2^3+5*g^1*2^4+1*g^0*2^5\).
03

Simplify the Binomial Expansion

Simplifying the above expression, one arrives at: \(g^5+10g^4+40g^3+80g^2+80g+32\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Binomial Expansion
Binomial expansion is a crucial concept in algebra that helps us expand expressions of the form \((a + b)^n\). It involves expanding such expressions using the Binomial Theorem, which provides a formula for expanding powers of binomials into a sum of terms. Here's how it works:
  • Identify the binomial expression you want to expand. For example, \((g + 2)^5\).
  • Determine the power, \(n\). In this case, \(n = 5\).
  • Use Pascal's Triangle to find the coefficients for each term in the expansion.
The expanded form for this expression relies on both the coefficients found using Pascal's Triangle and the decreasing powers of the first term coupled with increasing powers of the second term. This method streamlines the process of binomial expansion and provides a systematic way to solve complex algebraic expressions.
Role of Coefficients in Binomial Expansion
Coefficients are numerical or constant quantities placed before and multiplying the variable terms in an algebraic expression. In the context of binomial expansions, coefficients play a vital role in determining the size and shape of each term in the expansion. To find these coefficients quickly, we use Pascal's Triangle:
  • Start from 0 and count the rows of Pascal's Triangle until you reach the desired power of the binomial. Here, for \((g + 2)^5\), we need the 5th row.
  • The coefficients from the 5th row are \(1, 5, 10, 10, 5, 1\).
These coefficients are then used as multipliers for combinatorial expressions within the expanded form of the binomial. They help in balancing the expression as it states how many ways the terms can be combined or arranged.
Foundations of Algebra in Binomial Expansion
Algebra provides the foundation for understanding and manipulating symbols and formulas. In binomial expansion, algebraic fundamentals such as powers, terms, and simplification are key. Here's how it links with our example:
  • The expression \((g + 2)^5\) is an algebraic expression where we apply the rules of exponents.
  • During expansion, each term of the binomial is raised to a power and multiplied by the corresponding coefficient.
  • The final step involves simplifying the expanded terms, combining like terms, and using basic arithmetic operations.
Simplifying the expression involves multiplying out the powers and simplifying any arithmetic involved, which requires a solid grasp of algebraic principles. This process demonstrates the elegant relationship between different elements of algebra, resulting in a fully expanded form such as \(g^5 + 10g^4 + 40g^3 + 80g^2 + 80g + 32\).

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Most popular questions from this chapter

The volume (in cubic inches) of a rectangular birdcage can be modeled by \(V=3 x^3-17 x^2+29 x-15\), where \(x\) is the length (in inches). Determine the values of \(x\) for which the model makes sense. Explain your reasoning.

Graph the function. Identify the \(x\)-intercepts and the points where the local maximums and local minimums occur. Determine the intervals for which the function is increasing or decreasing. $$h(x)=x^5+2 x^2-17 x-4$$

Use finite differences to determine the degree of the polynomial function that fits the data. Then use technology to find the polynomial function.$$ \begin{array}{|l|c|c|c|c|c|c|} \hline \boldsymbol{x} & -6 & -3 & 0 & 3 & 6 & 9 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & -2 & 15 & -4 & 49 & 282 & 803 \\ \hline \end{array} $$

THOUGHT PROVOKING Write and graph a polynomial function of degree 5 that has all positive or negative real zeros. Label each \(x\)-intercept. Then write the function in standard form.

In Exercises 11鈥�18, divide using synthetic division. $$ \left(x^2+8 x+1\right) \div(x-4) $$
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