2. Pure Mathematics AS 2

If the other side contains $x$, locate every zero of both expressions, split the number line into sign regions, remove the modulus with the correct sign in each region, solve and intersect with that region.

Write every polynomial in descending powers and insert zero coefficients for missing powers. Divide leading terms, multiply the divisor, subtract the whole row, bring down the next term and repeat until the remainder degree is smaller than the divisor degree.

Translate every factor or stated remainder into a substitution equation. If coefficients are unknown, solve the resulting simultaneous equations; then divide out confirmed factors and solve any lower-degree quotient when roots are required.

For a non-zero remainder, keep the right side: if division by $x+2$ leaves $5$, then $P(-2)=5$, not $0$. Use polynomial division when the full quotient is needed.

Record the positivity conditions before combining or expanding logs. Move coefficients into powers when useful, combine to one logarithm, then translate back to index form.

For $k>0$, $e^{kx}$ increases; for $k<0$, it decreases. Both stay positive and approach, but never cross, the $x$-axis in one direction.

Rearrange until each exponential expression is positive, take $\ln$ of both sides, use $\ln(a^{g(x)})=g(x)\ln a$, then solve the resulting algebraic equation and check in the original.

If terms such as $a^{2x}$ and $a^x$ occur together, set $t=a^x$ with $t>0$, solve the polynomial in $t$, reject non-positive roots, then take logs to recover $x$.

Transform measured uncertainties and units consistently, then interpret the gradient and intercept in the original parameters. A straight plot supports the model but does not prove causation.

If ln y=0.7x+1.2, then a=e^{1.2} and b=e^{0.7} for y=ab^x.

Start with the corresponding cosine, sine or tangent graph over the required angles. Its denominator zeros become reciprocal asymptotes; where the denominator is $\pm1$, the reciprocal is also $\pm1$; use the denominator sign between asymptotes to choose each branch.

$\sec(\pi/3)=2$ because $\cos(\pi/3)=1/2$. At $x=\pi/2$, cosine is zero, so secant is undefined and has a vertical asymptote.

Write $a\sin heta+b\cos heta=R\sin( heta+\alpha)$ by matching $R\cos\alpha=a$ and $R\sin\alpha=b$. Thus $R=\sqrt{a^2+b^2}$ and choose the quadrant of $\alpha$ from both signs. A cosine form is equally valid when coefficient matching is adjusted.

For exact values, expand a compound angle built from known angles. For equations, rewrite to one function or $R$-form, use its range to decide existence, find all solutions in the stated interval, and check any denominator restrictions.

Identify the outer function, write its base derivative with the inner expression unchanged, multiply by the inner derivative, then combine constant multiples, sums and differences.

Label $u$ and $v$, compute $u'$ and $v'$ separately using chain rules where needed, substitute without changing the quotient numerator order, then factor or simplify.

Find the parameter or point coordinates first, evaluate the tangent gradient $m$, then use $y-y_0=m(x-x_0)$. A non-vertical normal has gradient $-1/m$; handle horizontal/vertical tangent cases geometrically.

Match the whole integrand to one row, keep the linear inner expression unchanged, divide by its gradient $a$, carry any outside constant, and include $+C$ for an indefinite integral.

Differentiate the answer: the chain factor $a$ must cancel the inserted $1/a$. For definite integrals, use an interval that does not cross a point where the integrand is undefined.

First rewrite the squared sine or cosine as a constant plus/minus a double-angle cosine. Then integrate term by term using the linear-inner rule, including the factor created by the doubled angle.

Confirm equal spacing, list all $n+1$ ordinates in order, weight endpoints once and internal ordinates twice, then retain appropriate accuracy because this is an estimate.

For a continuous $f$, $f(1)<0$ and $f(2)>0$ locates at least one root between $1$ and $2$. State the function values or their signs, not just the interval.

A graph gives visual approximate evidence; a sign-change bracket gives endpoint evidence. A narrower bracket gives a tighter location but remains an interval, not the exact root.

Choose a starting value in the stated interval, compute enough figures during iteration, and stop using a tolerance on successive values or the residual. Different rearrangements can converge differently.

For x=cos x, starting x₀=0 gives a sequence approaching about 0.739; starting values should remain in a region where g behaves stably.

Use the stated starting value, keep guard digits, tabulate $n$ and $x_n$, apply the same formula repeatedly, watch for settling/divergence/cycling, and continue until successive values justify the prescribed rounded answer.

For a requested number of decimal places, obtain successive values that round consistently at that precision and substitute the reported approximation into the original equation as a residual sense-check when practical.