CP-4 Further algebra and functions — 9FM0 spec coverage

CP-4.1

Understand and use the relationship between roots and coefficients of polynomial equations up to quartic equations.

For a monic cubic with roots $\alpha,\beta,\gamma$ , the coefficients give $\alpha+\beta+\gamma=-a$ , $\alpha\beta+\beta\gamma+\gamma\alpha=b$ and $\alpha\beta\gamma=-c$ in $x^3+ax^2+bx+c=0$ .
For a monic quartic, the elementary symmetric sums have alternating signs: sum of roots, pair products, triple products and the product are $-a,b,-c,d$ .
Derived sums are built from these relations; for example, $\sum\alpha^2=(\sum\alpha)^2-2\sum\alpha\beta$ .
A common error is to ignore the leading coefficient or lose the alternating signs when the polynomial is not monic.

Tier 1 · Easy

3 marks

ORIGINAL

The roots of $x^3-5x^2+2x+8=0$ are $\alpha,\beta,\gamma$ . Without solving the equation, find $\alpha+\beta+\gamma$ , $\alpha\beta+\beta\gamma+\gamma\alpha$ and $\alpha\beta\gamma$ .

Tier 2 · Standard

4 marks

ORIGINAL

The non-zero roots of $2x^4-3x^3-5x^2+7x-4=0$ are $\alpha,\beta,\gamma,\delta$ . Without solving the equation, find $\alpha+\beta+\gamma+\delta$ and $\frac1\alpha+\frac1\beta+\frac1\gamma+\frac1\delta$ .

Tier 3 · Hard

5 marks

ORIGINAL

The roots of $x^3-4x^2+px-6=0$ are $\alpha,\beta,\gamma$ , and $\alpha^2+\beta^2+\gamma^2=10$ . Determine $p$ and then find $\alpha^3+\beta^3+\gamma^3$ without solving the cubic.

CP-4.2

Form a polynomial equation whose roots are a linear transformation of the roots of a given polynomial equation (of at least cubic degree).

If a new root is $y=a\alpha+b$ , rearrange to $\alpha=(y-b)/a$ and substitute this expression into the original polynomial.
After substitution, multiply by a suitable power of $a$ to clear denominators, then collect powers of the new variable.
For a shift $y=\alpha+b$ , replace the old variable by $y-b$ ; for a scale $y=a\alpha$ , replace it by $y/a$ .
A common error is to substitute the forward transformation $a y+b$ instead of expressing the old root in terms of the new one.

Tier 1 · Easy

3 marks

ORIGINAL

The roots of $t^3-3t+1=0$ are $\alpha,\beta,\gamma$ . Form a polynomial equation whose roots are $\alpha+2,\beta+2,\gamma+2$ .

Tier 2 · Standard

4 marks

ORIGINAL

The roots of $2t^3-t^2+4t-3=0$ are $\alpha,\beta,\gamma$ . Form a polynomial equation whose roots are $3\alpha-1$ , $3\beta-1$ and $3\gamma-1$ .

Tier 3 · Hard

6 marks

ORIGINAL

The roots of $t^4-2t^3+t^2+3t-1=0$ are $\alpha,\beta,\gamma,\delta$ . Form a polynomial equation whose roots are $2-3\alpha$ , $2-3\beta$ , $2-3\gamma$ and $2-3\delta$ .

CP-4.3

Understand and use formulae for the sums of integers, squares and cubes and use these to sum other series.

Use $\sum_{r=1}^n r=\frac{n(n+1)}2$ , $\sum_{r=1}^n r^2=\frac{n(n+1)(2n+1)}6$ and $\sum_{r=1}^n r^3=\left[\frac{n(n+1)}2\right]^2$ .
Expand a polynomial expression in $r$ , split the summation term by term, and then substitute the standard formulae.
For example, $\sum(3r^2-2r+4)=3\sum r^2-2\sum r+4n$ before simplification.
A common error is to treat $\sum 1$ as $1$ instead of $n$ , or to change the summation limits while expanding.

Tier 1 · Easy

3 marks

ORIGINAL

Evaluate $\sum_{r=1}^{20}r(r+1)$ .

Tier 2 · Standard

4 marks

ORIGINAL

Show that $\sum_{r=1}^{n}(3r^2-2r+4)=\frac{n(2n^2+n+7)}2$ .

Tier 3 · Hard

6 marks

ORIGINAL

Prove that $\sum_{r=1}^{n}r(r+1)(2r+1)=\frac{n(n+1)^2(n+2)}2$ . Hence evaluate the sum when $n=15$ .

CP-4.4

Understand and use the method of differences for summation of series including use of partial fractions.

The method of differences rewrites a term as $f(r)-f(r+1)$ , so intermediate terms cancel when the series is written out.
Partial fractions often reveal the required difference, such as $\frac1{r(r+1)}=\frac1r-\frac1{r+1}$ .
Write the first few and last few terms before cancelling; the surviving boundary terms determine the finite sum.
A common error is to cancel the final terms as well as the intermediate ones or to omit a constant factor from the partial fractions.

Tier 1 · Easy

3 marks

ORIGINAL

Use the method of differences to find $\sum_{r=1}^{n}\frac1{(r+2)(r+3)}$ .

Tier 2 · Standard

5 marks

ORIGINAL

Find a closed form for $\sum_{r=1}^{n}\frac1{r(r+2)}$ . Hence evaluate the sum for $n=10$ .

Tier 3 · Hard

6 marks

ORIGINAL

Use partial fractions and the method of differences to prove that $\sum_{r=1}^{n}\frac1{r(r+1)(r+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$ . Hence find the corresponding infinite sum.

CP-4.5

Find the Maclaurin series of a function including the general term.

The Maclaurin series is $f(x)=\sum_{r=0}^{\infty}\frac{f^{(r)}(0)}{r!}x^r$ wherever the expansion is valid.
Differentiate repeatedly, identify $f^{(r)}(0)$ , and include factorials and powers of $x$ explicitly in the general term.
Products or composites can sometimes be handled more efficiently by combining known series or by using a complex exponential and taking a real part.
A common error is to give only the first few terms when the question asks for a general term, or to omit $r!$ from the denominator.

Tier 1 · Easy

3 marks

ORIGINAL

Find the Maclaurin series of $f(x)=\frac1{1-x}$ and give its general term.

Tier 2 · Standard

4 marks

ORIGINAL

Find the Maclaurin series of $e^{2x}$ through degree $4$ , and state the general term.

Tier 3 · Hard

6 marks

ORIGINAL

Find the Maclaurin series of $e^x\cos x$ through degree $5$ , and give a general term for the series.

CP-4.6

Recognise and use the Maclaurin series for e^x, ln(1+x), sin x, cos x and (1+x)^n, and be aware of the range of values of x for which they are valid (proof not required).

$e^x=\sum_{r=0}^{\infty}\frac{x^r}{r!}$ , while $\sin x=x-\frac{x^3}{3!}+\cdots$ and $\cos x=1-\frac{x^2}{2!}+\cdots$ ; all three are valid for every real $x$ .
$\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$ for $-1<x\leq1$ .
The binomial series $(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\cdots$ is valid for $|x|<1$ when it does not terminate; a non-negative integer power gives an identity for all $x$ .
A common error is to substitute $ax$ into a standard series without also changing its range of validity.

Tier 1 · Easy

3 marks

ORIGINAL

Write down the Maclaurin series for $\cos x$ through degree $6$ , and state its range of validity.

Tier 2 · Standard

4 marks

ORIGINAL

Write the first four terms of the expansion of $(1-2x)^{-1/2}$ , and state the range of values of $x$ for which it is valid.

Tier 3 · Hard

6 marks

ORIGINAL

Use standard Maclaurin series to show that $\ln\left(\frac{1+x}{1-x}\right)=2\left(x+\frac{x^3}{3}+\frac{x^5}{5}+\cdots\right)$ . State the range of validity and use terms up to $x^5$ to approximate $\ln(3/2)$ .