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Chapter 1: Problem 12

Express as a polynomial. $$(x+3 y)^{3}$$

Short Answer

Expert verified
The polynomial is \(x^3 + 9x^2 y + 27xy^2 + 27y^3\).

Step by step solution

01

Apply the Binomial Theorem

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \(a = x\), \(b = 3y\), and \(n = 3\). We will expand \((x + 3y)^3\) using this formula.
02

Calculate Each Term Independently

Substitute into the Binomial Theorem: \[(x + 3y)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} (3y)^k\]This gives us individual terms: - For \(k = 0\): \(\binom{3}{0} x^3 (3y)^0 = x^3\) - For \(k = 1\): \(\binom{3}{1} x^2 (3y)^1 = 3x^2 \cdot 3y = 9x^2 y\) - For \(k = 2\): \(\binom{3}{2} x^1 (3y)^2 = 3x \cdot 9y^2 = 27x y^2\) - For \(k = 3\): \(\binom{3}{3} x^0 (3y)^3 = 27y^3\)
03

Sum All Calculated Terms

Now, add all the terms from Step 2 to express the polynomial:\[x^3 + 9x^2 y + 27xy^2 + 27y^3\].This polynomial is the final expanded form of \((x + 3y)^3\).

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Key Concepts

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

Polynomial Expansion
Polynomial expansion is a fundamental concept in algebra. It involves transforming an expression with exponents into a series of terms. In our example, we expand \((x+3y)^{3}\) into a polynomial. This means we will rewrite it as a sum of multiple terms without exponents. Polynomial expansion is especially useful for simplifying expressions and solving equations.
  • The process uses rules and formulas, such as the Binomial Theorem, to break down complex expressions into simpler parts.
  • This transformation helps us to better understand the structure and function of algebraic expressions.
Expanding a polynomial allows us to see every contribution to the final form of the expression.
Binomial Coefficients
Binomial coefficients play a key role in expanding expressions like \((x+3y)^3\). These coefficients are the numbers that multiply the terms in the polynomial. They arise from the Binomial Theorem, represented by \(\binom{n}{k}\), where \(n\) is the total exponent and \(k\) is the specific term index.
  • The binomial coefficient tells us how many ways we can choose \(k\) items from \(n\) items, which directly influences the terms' multiplication factor during expansion.
  • For example, the coefficients for \((x+3y)^{3}\) are found from \(\binom{3}{k}\) for \(k = 0, 1, 2, 3\).
  • This results in coefficients 1, 3, 3, and 1, which are used to scale the corresponding terms \(x^{3-k} (3y)^k\).
Understanding binomial coefficients is essential as they determine the balance and symmetry in polynomial expressions.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They form the backbone of algebra and include terms like \(x, 3y,\) and \(x^3\). In our exercise, \((x+3y)^3\) is expanded into another algebraic expression: \(x^3 + 9x^2 y + 27xy^2 + 27y^3\).
  • Each term in the expression represents a distinct component that contributes to the expression's value.
  • By manipulating variables and coefficients, algebraic expressions can be added, subtracted, multiplied, and divided.
  • When expanding \((x+3y)^3\), each term, like \(9x^2y\), results from combining coefficients with powers of \(x\) and \(y\).
Grasping algebraic expressions equips you with the tools needed to construct, deconstruct, and simplify mathematical problems.

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Most popular questions from this chapter

A large grain silo is to be constructed in the shape of a circular cylinder with a hemisphere attached to the top (see the figure). The diameter of the silo is to be 30 feet, but the height is yet to be determined. Find the height \(h\) of the silo that will result in a capacity of \(11,250 \pi \mathrm{ft}^{3}\).

Rewrite the expression using a radical. (a) \(4+x^{3 / 2}\) \((b)(4+x)^{3 / 2}\)

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt{\frac{1}{7}}$$

In evaluating negative numbers raised to fractional powers, it may be necessary to evaluate the root and integer power separately. For example, \((-3)^{2 / 5}\) can be evaluated successfully as \(\left[(-3)^{15}\right]^{2}\) or \(\left[(-3)^{2}\right]^{1 / 3}\), whereas an error message might otherwise appear. Approximate the real number expression to four decimal places. (a) \((-3)^{2 / 5}\) (b) \((-5)^{4 / 3}\)

Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$\left(a^{2}+1\right)^{1 / 2} \square a+1$$
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