Linear
Product
Quadratic
Sum

Distributive Law

A product of factors can be converted to a sum of terms.

A(B + C) = AB + AC

examples:

2(5p - 6) = 2(5p) - 2(6) = 10p - 12

5p²(2p - 3) = 5p²(2p) - 3(5p²) = 10p³ - 15p²

Another very common expansion is:

(A + B)(C + D) = AC + AD + BC + BD

examples:

(5 - 2y)(y - 7) = 5(y) + 5(-7) - 2y(y) -2y(-7) = 5y - 35 - 2y² + 14y = -35 + 19y - 2y²

Exponent

For exponents which are natural numbers, the exponent gives the number of factors which are multiplied together.

examples

16 = 4 x 4 = 4² the exponent 2 indicates that two factors of 4 are multiplied.

x⁶ = (x)(x)(x)(x)(x)(x) The exponent 6 indicates that 6 factors x are multiplied together.

Factor

A number or expression is a factor of another number or expression if it can be MULTIPLIED by some number or expression (a factor) to obtain the other number or expression.

examples:

-6, -3, -2, -1, 1, 2, 3 and 6 are all factors of 6

-3, -1, 1, 3, (x + 2), (3x + 6), (-x - 2), (-3x-6) are all factors of 3x + 6 because

3(x + 2) = 3x + 6
-3(-x - 2) = 3x + 6
-1(-3x-6) = 3x + 6
1(3x + 6) = 3x + 6

x is not a factor of x + 4 because (x + 4) has 4 ADDED to x, not MULTIPLIED.

Highest Common Factor

For numbers this the largest number which is a factor of all the numbers in question.

examples:

2 is the HCF of 4 and 6
20 is the HCF of 380 and 140

For expressions this is the largest part of the expression which is in common to all the expressions in question.

examples:

5p²(2p - 3) is the HCF of 30p²(2p - 3) and 25p⁴(2p - 3)³

Linear

Linear expressions involve only the first power of a variable, not higher powers. When graphed, linear expressions give straight line graphs.

examples:

y + 2 is a linear expression
0.2x - 15
77 - 9z

Product

A product of terms is a number of terms which are MULTIPLIED together.

examples:

5x means 5 multiplied by x, that is, the product of 5 and x.
7x²e^-x means 7 multiplied by x² multiplied by e^-x, that is the product of 7, x² and e^-x
(2x + 8)⁶ means 6 factors, each (2x + 8) multiplied together, that is the product of 6 factors of (2x + 8)

Quadratics

Quadratic expressions (or simply "quadratics"), involving the square of a variable as well as the first power of the variable. In general, a quadratic expression has the form

Ax² + Bx + C

where A, B and C are numbers (which may be positive, negative or zero, except that A must not be zero). A is called the coefficient of x², B is called the coefficient of x and C is called the constant term.

example

4x² – 5x + 8 is a quadratic. Here the coefficient of x² is 4, the coefficient of x is –5 and the constant term is 8, so for this quadratic we have A = 4, B = –5 and C = 8.

Sum

A sum of terms is a number of terms which are ADDED together.

examples:

x + 6 means add 6 to x, that is the sum of x and 6.
5x - 2 means add -2 to 5x (as it can be expressed as 5x + -2), that is the sum of 5x and -2.