Algebra flashcards

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What is the piecewise definition of the modulus function, $|x|$?

$|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$. It gives the non-negative value of a number.

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What are the two main algebraic methods for solving an equation of the form $|ax+b| = c$ (where $c>0$)?

1. Square both sides: $(ax+b)^2 = c^2$. 2. Use the definition: $ax+b = c$ or $ax+b = -c$. Method 2 is usually faster.

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How do you solve an inequality like $|ax+b| < c$ (where $c>0$)?

Convert it to the double inequality $-c < ax+b < c$ and solve for $x$. For $|ax+b| > c$, you solve $ax+b > c$ or $ax+b < -c$.

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State the Factor Theorem.

If $P(x)$ is a polynomial and $P(a) = 0$, then $(x-a)$ is a factor of $P(x)$.

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State the Remainder Theorem.

When a polynomial $P(x)$ is divided by $(x-a)$, the remainder is $P(a)$.

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What is a 'proper' rational function?

A rational function $\frac{P(x)}{Q(x)}$ where the degree of the numerator polynomial $P(x)$ is less than the degree of the denominator polynomial $Q(x)$.

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What must you do first if you are asked to express an 'improper' rational function in partial fractions?

Perform polynomial long division to get a quotient polynomial and a proper rational function remainder, which can then be split into partial fractions.

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What is the general form of the binomial expansion for $(1+x)^n$ where $n$ is a rational number?

$1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots$

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What is the crucial condition for the binomial expansion of $(1+ax)^n$ to be valid?

The modulus of the term being expanded with '1' must be less than 1. For $(1+ax)^n$, the condition is $|ax| < 1$, which simplifies to $|x| < \frac{1}{|a|}$.

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What is the partial fraction form for $\frac{x+1}{(x-2)(x+3)^2}$?

$\frac{A}{x-2} + \frac{B}{x+3} + \frac{C}{(x+3)^2}$. A common mistake is to forget the $\frac{B}{x+3}$ term for the repeated root.