1. Aims: To understand what a polynomial is To be able to add and subtract polynomials To be able to multiply polynomials Polynomials/Factorisation Lesson 1

2. Polynomials A polynomial in x is an expression of the form where a, b, c, … are constant coefficients and n is a positive integer. Examples of polynomials include: Polynomials are usually written in d_____________________ powers of x. 3 x 7 + 4 x 3 – x + 8 x 11 – 2 x 8 + 9 x 5 + 3 x 2 – 2 x 3.and The value of a is called the leading c_____________________.

3. Polynomials A polynomial of degree 1 is called l_________ and has the general form ax + b. A polynomial of degree 2 is called q___________ and has the general form ax 2 + bx + c. A polynomial of degree 3 is called c_______ and has the general form ax 3 + bx 2 + cx + d. A polynomial of degree 4 is called q_________________ and has the general form ax 4 + bx 3 + cx 2 + dx + e. The degree, or order, of a polynomial is given by the highest power of the variable.

4. Adding and subtracting polynomials When two or more polynomials are added, subtracted or multiplied, the result is another polynomial. Find a) f ( x ) + g ( x ) b) 2 f ( x ) – g ( x ) Polynomials are added and subtracted by collecting like terms. For example: f ( x ) = 2 x 3 – 5 x + 4 and g ( x ) = 2 x – 4 Polynomials are often expressed using function notation.

5. Multiplying polynomials When two polynomials are multiplied together every term in the first polynomial must by multiplied by every term in the second polynomial. For example: f ( x ) = 3 x 3 – 2 and g ( x ) = x 3 + 5 x – 1 f ( x ) g ( x ) = ( 3 x 3 – 2 ) ( x 3 + 5 x – 1 )

6. Multiplying polynomials Sometimes we only need to find the coefficient of a single term. For example: Find the coefficient of x 2 when x 3 – 4 x 2 + 2 x is multiplied by 2 x 3 + 5 x 2 – x – 6. We don’t need to multiply this out in full. We only need to decide which terms will multiply together to give terms in x 2. ( x 3 – 4 x 2 + 2 x ) (2 x 3 + 5 x 2 – x – 6) We have: So, the coefficient of x 2 is ____.

7. Multiplying polynomials The product (Ax + B)(2x – 9) = 6 x 2 – 19 x – 36 where A and B are constants. Find the values of A and B (Ax + B)(2x – 9) = 6 x 2 – 19 x – 36 Multiply out the LHS Complete Treasure Hunt Do exercise 9 on page 135 qu 1,6,7 Now equate the

8. Challenge