9709 · 3.1
Algebra flashcards
Revision flashcards for Cambridge 9709 Algebra (syllabus 3.1). Flip, recall, then mark a real past-paper question.
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What are the two algebraic methods for solving an equation of the form $|f(x)| = g(x)$?
1. Consider two separate cases: $f(x) = g(x)$ and $f(x) = -g(x)$. 2. Square both sides to get $(f(x))^2 = (g(x))^2$. Always check for extraneous solutions.
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How do you sketch the graph of $y = |f(x)|$ from the graph of $y = f(x)$?
Reflect the part of the graph of $y=f(x)$ that is below the x-axis in the x-axis. The part of the graph already on or above the x-axis remains unchanged.
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What is the general form for partial fractions with a repeated linear factor in the denominator, e.g., $(x-a)^2$?
The expression will have terms $\frac{A}{x-a} + \frac{B}{(x-a)^2}$.
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What is the general form for partial fractions with an irreducible quadratic factor in the denominator, e.g., $(x^2+b)$?
The corresponding term in the partial fraction decomposition is $\frac{Ax+B}{x^2+b}$.
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What is the binomial expansion of $(1+x)^n$ for a rational number $n$?
$1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots$
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What is the condition for the validity of the binomial expansion of $(a+bx)^n$?
The expansion is valid when $|\frac{bx}{a}| < 1$, which simplifies to $|x| < |\frac{a}{b}|$.
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State the Factor Theorem.
For a polynomial $P(x)$, if $P(a) = 0$, 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 $R = P(a)$.
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What is the Complex Conjugate Root Theorem?
If a polynomial has real coefficients, and $z = a+bi$ is a root, then its complex conjugate $z^* = a-bi$ must also be a root.
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What real quadratic factor corresponds to the complex conjugate roots $a \pm bi$?
The factor is $(x-(a+bi))(x-(a-bi)) = x^2 - 2ax + (a^2+b^2)$.
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Trap: When solving $|x-5| < 2$, what is a common error?
A common error is to only solve $x-5 < 2$ to get $x<7$. You must also consider $-(x-5) < 2$, which gives $-x+5 < 2$, so $3 < x$. The full solution is $3 < x < 7$.
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Trap: When expanding $(4-x)^{1/2}$, what is the first step?
You must first factor out the 4 to match the $(1+y)^n$ form: $4^{1/2}(1 - \frac{x}{4})^{1/2} = 2(1 - \frac{x}{4})^{1/2}$. Forgetting to apply the power to the factored-out constant is a common mistake.