Algebraic methods

Question 1

when simplifying an algebraic fraction what should you do?

Answer 1

factorise both numerator and denominator, and then cancel common factors.

Question 2

a polynomial with examples

Answer 2

finite expression with positive whole number indices:
2x + 4 ✓
4xy² + 3x - 9 ✓
8 ✓
√x ✗
6x-² ✗
4/x ✗

Question 3

factor theorem

Answer 3

if f(x) is a polynomial, then:
• if f(p) = 0, (x - p) is a factor of f(x)
• if (x - p) is a factor of f(x), then f(p) = 0

Question 4

dividing polynomials long division

Answer 4

can divide by (x ± p)

Question 5

ways to prove a mathematical statement

Answer 5

• deduction
• exhaustion
• counter example

Question 6

deduction

Answer 6

start from known facts/definitions and use logical steps to reach the desired conclusion

Question 7

exhaustion

Answer 7

breaking the statement into smaller cases and proving each case separately

Question 8

counter-example

Answer 8

give an example that does not work for the statement, one is sufficient

Question 9

in a mathematical proof you must

Answer 9

• state any information or assumptions used
• show every step of proof clearly
• make sure each step follows logically from previous step
• make sure all possible cases are covered
• write a statement of proof at the end

Question 10

to prove an identity you should

Answer 10

• start with the expression on one side of the identity
• manipulate that expression algebraically until it matches the other side
• show every step of algebraic working