True or False? The binomial theorem only applies to linear expressions.

False. The binomial theorem applies to any two-term expression, but in IB it is most often applied to linear expressions.

True or False? Binomial coefficients are always integers .

True. Binomial coefficients are always integers .

True or False? In Pascal's triangle , each number is the sum of the two numbers directly above it.

True. In Pascal's triangle , each number is the sum of the two numbers directly above it.

What does the ellipsis (...) indicate in a binomial expansion?

The ellipsis (...) in a binomial expansion indicates that the expansion continues.

What does ' in ascending powers ' mean?

' In ascending powers ' means that the terms are arranged so that the power of the variable increases with each term.

True or False? The binomial theorem can be applied to negative and fractional indices .

True. The binomial theorem can be applied to negative and fractional indices .

True or False? The more terms used in a binomial expansion approximation, the closer the approximation is to the true value .

True. The more terms used in a binomial expansion approximation, the closer the approximation is to the true value .

IBMathsDPAnalysis & Approaches (AA)HLFlashcards1. Number & AlgebraBinomial Theorem

Binomial Theorem (DP IB Analysis & Approaches (AA))

Flashcards

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FrontBinomial Theorem
What is the binomial theorem?
FrontBinomial Theorem
State the equation for the binomial theorem.
BackBinomial Theorem
The equation for the binomial theorem is ${(a + b)}^{n} = a^{n} + {}^{n}{C_{1}} a^{n - 1} b + . . . + {}^{n}{C_{r}} a^{n - r} b^{r} + . . . + b^{n}$
Where:
${}^{n}{C_{r}} = \frac{n!}{r! (n - r)!}$
This equation is valid for any $n \in ℕ$ (i.e., $n = 0, 1, 2, 3, . . .$ ).
The equation is in the exam formula booklet.

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Cards in this collection (18)

What is the binomial theorem?
The binomial theorem is a method for expanding a two-term expression in a bracket raised to a power, e.g. ${(a + b)}^{n}$ .
True or False?
The binomial theorem only applies to linear expressions.
False.
The binomial theorem applies to any two-term expression, but in IB it is most often applied to linear expressions.
State the equation for the binomial theorem.
The equation for the binomial theorem is ${(a + b)}^{n} = a^{n} + {}^{n}{C_{1}} a^{n - 1} b + . . . + {}^{n}{C_{r}} a^{n - r} b^{r} + . . . + b^{n}$
Where:
- ${}^{n}{C_{r}} = \frac{n!}{r! (n - r)!}$
This equation is valid for any $n \in ℕ$ (i.e., $n = 0, 1, 2, 3, . . .$ ).
The equation is in the exam formula booklet.
What is the binomial coefficient?
The binomial coefficient ${}^{n}{C_{r}}$ is used to find the coefficients in a binomial expansion.
Its value is given by ${}^{n}{C_{r}} = \frac{n!}{r! (n - r)!}$ , which is in the exam formula booklet (although you will usually use your GDC to find the value of the coefficients in an expansion).
${}^{n}{C_{r}}$ also represents the number of ways to choose $r$ items out of $n$ different items.
True or False?
Binomial coefficients are always integers.
True.
Binomial coefficients are always integers.
True or False?
Pascal's triangle can be used to find binomial coefficients ${}^{n}{C_{r}}$ .
True.
Pascal's triangle is a triangular array of the binomial coefficients, and can be used to find ${}^{n}{C_{r}}$ for different values of $n$ and $r$ .
However Pascal's triangle becomes awkward to use when $n$ gets large.
True or False?
In Pascal's triangle, each number is the sum of the two numbers directly above it.
True.
In Pascal's triangle, each number is the sum of the two numbers directly above it.
True or False?
${}^{n}{C_{r}} = {}^{n}{C_{n - r}}$
True.
${}^{n}{C_{r}} = {}^{n}{C_{n - r}}$
E.g. ${}^{7}{C_{2}} = {}^{7}{C_{5}}$
What does the ellipsis (...) indicate in a binomial expansion?
The ellipsis (...) in a binomial expansion indicates that the expansion continues.
What does 'in ascending powers' mean?
'In ascending powers' means that the terms are arranged so that the power of the variable increases with each term.
True or False?
The binomial theorem can be applied to negative and fractional indices.
True.
The binomial theorem can be applied to negative and fractional indices.
What is the general form of the binomial expansion for ${(a + b)}^{n}$ where $n \in ℚ$ ?
The general form of the binomial expansion for ${(a + b)}^{n}$ , where $n \in ℚ$ , is ${(a + b)}^{n} = a^{n} (1 + n (\frac{b}{a}) + \frac{n (n - 1)}{2!} {(\frac{b}{a})}^{2} + \dots)$ .
This is in the exam formula booklet.
True or False?
All binomial expansions of ${(a + b)}^{n}$ end after a finite number of terms.
False.
Not all binomial expansions of ${(a + b)}^{n}$ end after a finite number of terms.
If $n$ is a positive integer (or zero), then the expansion will end after a finite number of terms.
If $n \in ℚ$ is not a positive integer (or zero), then the binomial expansion will have an infinite number of terms (it will'go on forever').
What is the general form of the binomial expansion for ${(1 + x)}^{n}$ where $n \in ℚ$ ?
The general form of the binomial expansion for ${(1 + x)}^{n}$ , where $n \in ℚ$ , is ${(1 + x)}^{n} = 1 + n x + \frac{n (n - 1)}{2!} x^{2} + \frac{n (n - 1) (n - 2)}{3!} x^{3} + \dots$ .
The interval of convergence for the expansion is $|x| < 1$ .
This formula is not in the exam formula booklet, but it can be derived from the formula for ${(a + b)}^{n}$ which is in the booklet.
What does "the interval of convergence" mean for a binomial expansion?
The interval of convergence is the range of x values for which a binomial expansion is valid.
E.g. the interval of convergence for $\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + . . .$ is $|x| < 1$ (which can also be written as $- 1 < x < 1$ ).
What is the interval of convergence for a binomial expansion of ${(a + b x)}^{n}$ .
The interval of convergence for a binomial expansion of ${(a + b x)}^{n}$ is $- \frac{a}{b} < x < \frac{a}{b}$ .
If you are using the binomial expansion ${(1 - x)}^{\frac{1}{2}} = 1 - \frac{x}{2} - \frac{x^{2}}{8} - \frac{x^{3}}{16} - \dots$ to approximate the value of $\sqrt{0.97}$ , what value of $x$ should you substitute into the expansion?
If you are using the binomial expansion ${(1 - x)}^{\frac{1}{2}} = 1 - \frac{x}{2} - \frac{x^{2}}{8} - \frac{x^{3}}{16} - \dots$ to approximate the value of $\sqrt{0.97}$ , you should substitute the value $x = 0.03$ into the expansion.
Compare the expressions: ${(1 - x)}^{\frac{1}{2}} = {(0.97)}^{\frac{1}{2}} \Rightarrow 1 - x = 0.97 \Rightarrow x = 0.03$ .
True or False?
The more terms used in a binomial expansion approximation, the closer the approximation is to the true value.
True.
The more terms used in a binomial expansion approximation, the closer the approximation is to the true value.

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