Reference Sheet: Programme F.2 — Introduction to Algebra

Rules, procedures and drill, in the order of the book's programme. Every rule links to the unit in the lesson that derives it — if a rule here looks arbitrary, the link is where it stops being arbitrary.

1. Letters as numbers

From Unit 01.

A letter stands for a number and is manipulated exactly like a numeral: a+a+a+a=4a, 3a-a=2a, 8a\div a=8, a\times a\times a\times a\times a=a^5.

Name	Written	Note
Sum	a+b
Difference	a-b
Product	a\times b=a.b=ab	the multiplication sign is usually suppressed
Quotient	a\div b=a/b=\frac ab	provided b\ne0
Power	a^b	a raised to the power b

Example. Think of a number a, add 15, double, add a, divide by 3, subtract a: a\to a+15\to2a+30\to3a+30\to a+10\to10, whatever a was.

2. Constants and variables

From Unit 01.

A constant stands for a single number, possibly unknown. A variable may stand for any one of a collection of numbers. Convention: a,b,c,d,\ldots for constants, \ldots,w,x,y,z for variables — but say which is which, because the role comes from the problem.

3. Rules of algebra

From Unit 01 and Unit 02.

Rule	Holds	Fails
Commutativity	x+y=y+x, xy=yx	x-y\ne y-x unless x=y; \frac xy\ne\frac yx unless x=y\ne0
Associativity	x+(y+z)=(x+y)+z, x(yz)=(xy)z	x-(y-z)\ne(x-y)-z unless z=0; x\div(y\div z)\ne(x\div y)\div z unless z=1
Distributivity	x(y\pm z)=xy\pm xz, (x\pm y)z=xz\pm yz	—
Division	from the right: \frac{x+y}z=\frac xz+\frac yz	from the left: \frac x{y+z}\ne\frac xy+\frac xz

4. Rules of precedence

From Unit 01.

Unchanged from arithmetic: brackets, then powers, then \times and \div from left to right, then + and - from left to right.

5. Terms, coefficients and like terms

From Unit 01.

Terms are separated by + and -. The coefficient is the number multiplying a term, including its sign: in 8x-3xy the coefficients are 8 and -3. Like terms contain identical letters and are collected by adding coefficients. Write the letters of a term in alphabetical order (yx\to xy).

Examples.

4x+3y-2z+5y-3x+4z=x+8y+2z

4uv-7uz-6wz+2uv+3wz=6uv-7uz-3wz

6. Similar terms

From Unit 01 and Unit 07.

Similar terms share some letters but not all. The shared part is a common factor and can be factored out:

ab+ac=a(b+c),\qquad 3pq-3qr=3q(p-r),\qquad 9st-3sv-6sw=3s(3t-v-2w).

7. Expanding brackets

From Unit 02.

Multiply (or divide) every term inside by the term outside. A negative term outside changes the sign of every term inside.

Examples.

3x(y-2z)=3xy-6xz,\qquad -2y(2x-4z)=-4xy+8yz

\frac{y+x}{8x}-\frac{y-x}{4x}=\frac y{8x}+\frac18-\frac y{4x}+\frac14=\frac{3x-y}{8x}

(2x-3)(x+4)=2x^2+8x-3x-12=2x^2+5x-12

8. Nested brackets

From Unit 02.

Remove the innermost brackets first.

Examples.

7(a-[4-5(b-3a)])=7(a-[4-5b+15a])=7(a-4+5b-15a)=35b-98a-28

4(2x+3[5-2(x-y)])=4(2x+3[5-2x+2y])=4(2x+15-6x+6y)=24y-16x+60

9. Powers and the rules of indices

From Unit 03.

In 5a^3: base a, index (exponent, power) 3, coefficient 5.

No.	Rule
1	a^m\times a^n=a^{m+n}
2	a^m\div a^n=a^{m-n}
3	(a^m)^n=a^{mn}
4	a^0=1 (a\ne0)
5	a^{-m}=\frac1{a^m} (a\ne0)
6	a^{1/m}=\sqrt[m]a

Examples.

\frac{x^7y^{-2}}{x^3y^{-5}}=x^4y^3,\qquad \frac{6x^{-4}\times2x^3}{8x^{-3}}=\frac32x^2

10. Fractional indices

From Unit 03.

a^{n/m}=\sqrt[m]{a^n}=\left(\sqrt[m]a\right)^n

Take the root first when the numbers are large: 16^{3/4}=(\sqrt[4]{16})^3=2^3=8. For a>0 the root is the positive one.

Example.

\left(5x^2y^{-3/2}z^{1/4}\right)^2\times\left(4x^4y^2z\right)^{-1/2}=25x^4y^{-3}z^{1/2}\times\frac12x^{-2}y^{-1}z^{-1/2}=\frac{25x^2}{2y^4}

11. Definition of the logarithm

From Unit 04.

If a=b^c with a>0, b>0, b\ne1, then c=\log_ba, "the log to the base b of a".

Examples. \log_525=2; \log_216=x\Rightarrow2^x=2^4, x=4; \log_x81=4\Rightarrow x^4=3^4, x=3; \log_7x=2\Rightarrow x=49.

12. Rules of logarithms

From Unit 04.

	Rule	In words
(a)	\log_axy=\log_ax+\log_ay	log of a product = sum of the logs
(b)	\log_a(x\div y)=\log_ax-\log_ay	log of a quotient = difference of the logs
(c)	\log_a(x^n)=n\log_ax	log of a power = power times the log
(d)	\log_a1=0
(e)	\log_aa=1
(f)	\log_aa^x=x
(g)	a^{\log_ax}=x
(h)	\log_ab=\frac1{\log_ba}

Examples. \log_327=3; 12^{\log_{12}4}=4; \frac1{\log_34}=\log_43; \log_a(5.889^{1.2})=1.2\log_a5.889.

13. Common and natural logarithms

From Unit 04.

Common logarithms, base 10, written \log. Natural logarithms, base e=2.71828\ldots, written \ln.

Argument	Log (base >1)	Example
1	0	\log1=0
greater than 1	positive	\log5.321=0.726, \ln13.45=2.599
between 0 and 1	negative	\log0.278=-0.556, \ln0.278=-1.280
0 or negative	not a real number	\ln(-0.001): calculator error

14. Change of base

From Unit 04.

The book's change of base formula:

\log_ba\,\log_ax=\log_bx,\qquad\text{so}\qquad \log_ax=\frac{\log x}{\log a}.

Examples. \log_23.66=\frac{0.5635}{0.3010}=1.872; \log_23.16=1.660; \ln0.278=\ln10\times\log0.278=2.303\times(-0.556)=-1.280.

15. Logarithmic equations

From Unit 04.

Collect each side into one logarithm using (a)–(c), equate the arguments, solve, then check that every original argument is positive.

Examples.

\log_ax^2+3\log_ax-2\log_a4x=\log_a\frac{x^2\cdot x^3}{16x^2}=\log_a\frac{x^3}{16}

2\log_ax-\log_a(x-1)=\log_a(x+3)\;\Rightarrow\;\frac{x^2}{x-1}=x+3\;\Rightarrow\;x=\frac32\quad(\text{valid: } x-1=\frac12>0)

16. Formulas in log form

From Unit 04.

Take logs of both sides and expand with (a)–(c); reverse the steps to remove logs.

Examples.

T=2\pi\sqrt{\frac lg}\;\Rightarrow\;\log T=\log2\pi+\frac12\log l-\frac12\log g

\log K=\log P-\log T+1.3\log V\;\Rightarrow\;K=\frac{PV^{1.3}}T,\qquad \ln A=\ln P+rn\;\Rightarrow\;A=Pe^{rn}

17. Multiplication

From Unit 05 and Unit 02.

Multiply the second expression by each term of the first and add, keeping each power of the variable in its own column. Insert a missing power with a zero coefficient.

Examples.

(2x+5)(x^2+3x+4)=2x^3+11x^2+23x+20

(2x+6)(4x^3+0x^2-5x-7)=8x^4+24x^3-10x^2-44x-42

Check by putting x=10: 25\times134=3350.

18. Division

From Unit 05.

Long division: divide the leading terms, multiply the divisor by the result, subtract, bring down the next term, repeat. Insert any missing power with a zero coefficient. The result is the quotient.

Examples.

\frac{12x^3-2x^2-3x+28}{3x+4}=4x^2-6x+7,\qquad \frac{4x^3+0x^2+13x+33}{2x+3}=2x^2-3x+11

\frac{2r^3+5r^2-4r-3}{r^2+2r-3}=2r+1,\qquad \frac{q^3+27}{q+3}=q^2-3q+9,\qquad \frac{a^3+8b^3}{a+2b}=a^2-2ab+4b^2

Check by putting the variable equal to 10: 4163\div23=181.

19. Adding and subtracting algebraic fractions

From Unit 06.

Use the LCM of the denominators:

\frac ab+\frac cd=\frac{ad+cb}{bd}\qquad(b\ne0,\ d\ne0)

Examples. \frac45+\frac37=\frac{43}{35}; \frac ab-\frac c{d^2}+\frac da=\frac{a^2d^2-abc+bd^3}{abd^2}; \frac2{x+1}+\frac4{x+2}=\frac{6x+8}{x^2+3x+2}.

20. Multiplying and dividing algebraic fractions

From Unit 06.

\frac ab\times\frac cd=\frac{ac}{bd},\qquad \frac ab\div\frac cd=\frac ab\times\frac dc=\frac{ad}{bc}

The reciprocal of \frac cd is \frac dc. Work \times and \div left to right.

Examples. \frac{2a}{3b}\div\frac{a^2b}6=\frac4{ab^2}; \frac{2a}{3b}\div\frac{a^2b}6\times\frac{ab}2=\frac4{ab^2}\times\frac{ab}2=\frac2b.

21. Simplifying a fraction by factorizing

From Unit 06 and Unit 07.

Factorize numerator and denominator, then cancel common factors. Note the values excluded by the original denominator.

Examples.

\frac{25ab^2-15a^2b}{40ab^2-24a^2b}=\frac{5ab(5b-3a)}{8ab(5b-3a)}=\frac58

\frac{x^2-4}{x^2-x-2}=\frac{(x-2)(x+2)}{(x-2)(x+1)}=\frac{x+2}{x+1}\qquad(x\ne2,\ x\ne-1)

22. Common factors

From Unit 07.

Take out the HCF: the HCF of the coefficients, times the lowest power of each letter that appears in every term.

Examples. 10x+8=2(5x+4); 35x^2y^2-10xy^3=5xy^2(7x-2y); 8x^4y^3+6x^3y^2=2x^3y^2(4xy+3).

23. Common factors by grouping

From Unit 07.

Pair the four terms, factor each pair, then take out the common bracket. If no common bracket appears, rearrange the terms and try again.

Examples.

x^3-4x^2y+xy^2-4y^3=x^2(x-4y)+y^2(x-4y)=(x-4y)(x^2+y^2)

20x^2-3y^2+4xy^2-15x=(20x^2-15x)+(4xy^2-3y^2)=5x(4x-3)+y^2(4x-3)=(4x-3)(5x+y^2)

24. Useful products of two simple factors

From Unit 07 and Unit 02.

(a+b)^2=a^2+2ab+b^2,\qquad (a-b)^2=a^2-2ab+b^2,\qquad (a-b)(a+b)=a^2-b^2

The last is the difference of two squares. To recognize a perfect square, check that the middle term is twice the product of the square roots of the end terms.

Examples. x^2+10x+25=(x+5)^2; 4a^2-12a+9=(2a-3)^2; 16x^2+40xy+25y^2=(4x+5y)^2; 25x^2-16y^2=(5x-4y)(5x+4y); (2x+3y)^2-(x-4y)^2=(x+7y)(3x-y).

25. Quadratic expressions as the product of two factors

From Unit 07.

With a, b natural numbers:

Pattern	Constant term	Find a,b with
(x+a)(x+b)=x^2+(a+b)x+ab	positive	sum = coefficient of x, product = constant
(x-a)(x-b)=x^2-(a+b)x+ab	positive	sum = minus the coefficient of x
(x+a)(x-b)=x^2+(a-b)x-ab	negative	difference = coefficient of x

List the pairs with the right product, then pick the pair with the right sum or difference.

Examples. x^2+7x+12=(x+3)(x+4); x^2-11x+28=(x-4)(x-7); x^2-3x-18=(x-6)(x+3).

Leading coefficient not 1 (a method the book's exercises need but its text leaves out): for px^2+qx+r find two numbers with product pr and sum q, split the middle term with them, and group.

3x^2-11x-4:\ pr=-12,\ \text{pair } -12,\,1:\quad 3x^2-12x+x-4=3x(x-4)+(x-4)=(3x+1)(x-4)

Traps

Dividing by a sum. \frac x{y+z}\ne\frac xy+\frac xz: \frac{12}{3+1}=3 but \frac{12}3+\frac{12}1=16. Only the numerator splits.
Dropping the sign of a coefficient. In 8x-3xy the coefficient of xy is -3, not 3.
A minus sign before a bracket multiplies every term inside: -(x-3)=-x+3, not -x-3.
(a+b)^2\ne a^2+b^2. At a=3, b=4: 49 against 25. The two ab cells are missing.
\log(x+y)\ne\log x+\log y. \log(2+8)=1 but \log2+\log8=1.204. Rule (a) is about products.
Logs of zero or negatives. Not real numbers. Always check a log equation's answer against every original argument.
Multiplying before dividing. \times and \div run left to right; the wrong order in \frac{2a}{3b}\div\frac{a^2b}6\times\frac{ab}2 gives \frac8{a^2b^3} instead of \frac2b.
Forgetting the zero placeholder. Dividing 4x^3+13x+33 without 0x^2 puts the x terms in the x^2 column.
Cancelling terms instead of factors. \frac{x+2}{x}\ne2; only a factor of the whole numerator and the whole denominator cancels.
Losing excluded values when cancelling. They come from the original denominator.

Self-check

Simplify 5ab-3ba+2ac-ca+4bc.
Remove the brackets: 3(2x-[x-2(4-x)]).
Simplify (2a^3b^{-2})^3\div(4a^4b^{-2})^{1/2}.
Evaluate 27^{-2/3} without a calculator.
Evaluate to 3 dp: \log0.0425, \ln7.38, \log_320.
Solve \log_a(x+2)+\log_ax=\log_a3.
Express without logs: \log y=2\log x-\frac12\log(x+1)+\log5.
Multiply (3x-2)(x^3-4x+5).
Divide 2x^3-7x^2+9 by x-3.
Simplify \frac3{x-2}-\frac2{x+1} and state the excluded values.
Simplify \frac{3p}{4q}\div\frac{9p^2}{2q}\times\frac{pq}3.
Factorize: (a) 6x^2-x-12; (b) 9a^2-(a-2b)^2; (c) xy-3y+2x-6.

Fully worked solutions

1. ba=ab and ca=ac, so (5-3)ab+(2-1)ac+4bc=2ab+ac+4bc.

2. Innermost first: x-2(4-x)=x-8+2x=3x-8. Then 2x-(3x-8)=-x+8, and 3(-x+8)=24-3x. Check at x=5: 3(10-[5-2(-1)])=3(10-7)=9, and 24-15=9.

3. (2a^3b^{-2})^3=8a^9b^{-6} and (4a^4b^{-2})^{1/2}=2a^2b^{-1}. Dividing: 4a^{9-2}b^{-6-(-1)}=4a^7b^{-5}=\frac{4a^7}{b^5}.

4. 27^{-2/3}=\frac1{27^{2/3}}=\frac1{(\sqrt[3]{27})^2}=\frac1{3^2}=\frac19.

5. \log0.0425=-1.372 (negative, since 0.0425<1). \ln7.38=1.999. \log_320=\frac{\log20}{\log3}=\frac{1.3010}{0.4771}=2.727.

6. \log_ax(x+2)=\log_a3, so x^2+2x-3=0, (x+3)(x-1)=0, x=1 or x=-3. At x=-3 the term \log_ax has a negative argument, so it is rejected. x=1.

7. \log y=\log x^2-\log(x+1)^{1/2}+\log5=\log\frac{5x^2}{\sqrt{x+1}}, so y=\frac{5x^2}{\sqrt{x+1}}.

8. Insert the placeholder: x^3+0x^2-4x+5.

3x(x^3+0x^2-4x+5)=3x^4+0x^3-12x^2+15x

-2(x^3+0x^2-4x+5)=-2x^3+0x^2+8x-10

Adding the columns: 3x^4-2x^3-12x^2+23x-10.

9. Write 2x^3-7x^2+0x+9. 2x^3\div x=2x^2; subtract 2x^3-6x^2 to leave -x^2+0x. -x^2\div x=-x; subtract -x^2+3x to leave -3x+9. -3x\div x=-3; subtract -3x+9 to leave 0. Quotient 2x^2-x-3. Check at x=10: 1309\div7=187=200-10-3.

10. Excluded: x\ne2, x\ne-1.

\frac{3(x+1)-2(x-2)}{(x-2)(x+1)}=\frac{x+7}{(x-2)(x+1)}=\frac{x+7}{x^2-x-2}.

11. Left to right: \frac{3p}{4q}\times\frac{2q}{9p^2}=\frac{6pq}{36p^2q}=\frac1{6p}, then \frac1{6p}\times\frac{pq}3=\frac q{18}.

12. (a) Product 6\times(-12)=-72, sum -1: the pair is -9 and 8. 6x^2-9x+8x-12=3x(2x-3)+4(2x-3)=(2x-3)(3x+4).

(b) A difference of two squares with 3a and (a-2b): [3a-(a-2b)][3a+(a-2b)]=(2a+2b)(4a-2b)=4(a+b)(2a-b).