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Chapter 12: Problem 23

p(z)=4 z^{3}-z. Find the given values and simplify if possible. $$ p(t+1) $$

Short Answer

Expert verified
Answer: The simplified expression for $$p(t+1)$$ is $$4t^3 + 12t^2 + 11t + 3$$.

Step by step solution

01

Substitute t+1 for z

Replace $$z$$ in the polynomial function with $$(t+1)$$: $$ p(t+1) = 4(t+1)^3 - (t+1) $$
02

Expand (t+1)^3

Use the binomial theorem to expand $$(t+1)^3$$: $$ (t+1)^3 = t^3 + 3t^2 + 3t + 1 $$
03

Substitute the expansion into the polynomial function

Now, replace $$(t+1)^3$$ in $$p(t+1)$$ with the expanded expression: $$ p(t+1) = 4(t^3 + 3t^2 + 3t + 1) - (t+1) $$
04

Distribute and simplify

Distribute the $$4$$ in the expression and simplify by combining like terms: $$ p(t+1) = 4t^3 + 12t^2 + 12t + 4 - t - 1 $$ Combine the like terms: $$ p(t+1) = 4t^3 + 12t^2 + 11t + 3 $$ The simplified expression for $$p(t+1)$$ is $$4t^3 + 12t^2 + 11t + 3$$.

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Key Concepts

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

Binomial Theorem
The Binomial Theorem is a fundamental principle used to expand expressions raised to a power, particularly those of the form \((a + b)^n\). It provides a formula to expand these expressions without manually multiplying them out step-by-step. For any positive integer \(n\), the binomial theorem states that:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Where \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).
  • In our example, we expand \((t+1)^3\). According to the theorem, it becomes \(t^3 + 3t^2 + 3t + 1\).
  • This takes the form where \(a = t\), \(b = 1\), and \(n = 3\).
  • The coefficients for each term (1, 3, 3, 1) are derived from Pascal's triangle, which is a handy way to find binomial coefficients.
Understanding the binomial theorem makes polynomial expansion efficient, especially for higher powers.
Polynomial Expansion
Polynomial expansion refers to the process of simplifying expressions where variables are raised to powers, especially when multiplied together. This is a direct application from algebra that, without shortcuts like the binomial theorem, could be time-consuming.
  • In our case, the expression \((t+1)^3\) needs to be expanded before substituting back into the polynomial \(p(z) = 4z^3 - z\).
  • Using the binomial theorem, \((t+1)^3\) was expanded to \(t^3 + 3t^2 + 3t + 1\).
Now, the expanded expression allows you to substitute it into the function for a more manageable form to work with.
By expanding polynomials properly, further calculations such as distribution and combining like terms become clearer and more straightforward.
Combining Like Terms
Combining like terms is a basic yet crucial skill in simplifying expressions in algebra. It involves grouping terms with the same variable raised to the same power and simplifying them into a single term. For example:
  • In the function \(p(t+1) = 4(t^3 + 3t^2 + 3t + 1) - (t+1)\), after expanding and distributing, you obtain terms such as \(4t^3\), \(12t^2\), \(12t\), \(4\), \(-t\), and \(-1\).
  • Combine \(12t\) and \(-t\) to get \(11t\).
  • Similarly, \(4\) and \(-1\) combine to make \(3\).
This step transforms a complex expression into something much more straightforward: \(4t^3 + 12t^2 + 11t + 3\).

Combining like terms simplifies expressions, reducing them to their essential components and making them easier to understand and evaluate.

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Most popular questions from this chapter

$$ \begin{aligned} &\text { Find the solutions of }\\\ &\left(x^{2}-a^{2}\right)(x+1)=0, \quad a \text { a constant } \end{aligned} $$

Give the value of \(a\) that makes the statement true. The degree of \((t-1)^{3}+a(t+1)^{3}\) is less than 3

Without solving the equation, decide how many solutions it has. $$ (x-1)(x-2)=0 $$

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ 3-2(x-5)^{2} $$

Find a polynomial of degree 4 that has zeros at \(x=\) \(-2, x=-1, x=2,\) and \(x=3\) and whose graph contains the point (0,6) .
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