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“Teach A Level Maths” Vol. 1: AS Core Modules 10: Polynomials © Christine Crisp Polynomials Module C1 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Polynomials Polynomial Functions The following are examples of Polynomial Functions: x x3 2 x 3 2 x 2 13 x 10 x x 13 x 1 4 2 A quadratic polynomial A cubic polynomial A quartic polynomial Polynomials only contain terms of the type ax n, where n is a positive integer Polynomials Add 2x3 + 3x2 – 4 and x2 – x -2. Polynomials Find the x and x2 terms in the product of (2x + 3) (3x2 -2x +4). Polynomials If (Ax + B) (2x – 9) = (6x2 - 19x - 36), find A and B. Polynomials Expanding Cubic Functions e.g. 1 ( x 5)( x 2)( x 1) (x 2 2 x 5 x 10)( x 1) We multiply 2 of the parts together first, leaving the third unchanged Polynomials Expanding Cubic Functions e.g. 1 ( x 5)( x 2)( x 1) (x 2 2 x 5 x 10)( x 1) ( x 2 3 x 10)( x 1) We multiply 2 of the parts together first, leaving the third unchanged Now multiply each of the 3 terms in the 1st pair of brackets by each of the 2 terms in the 2nd Polynomials Expanding Cubic Functions e.g. 1 ( x 5)( x 2)( x 1) (x 2 2 x 5 x 10)( x 1) ( x 2 3 x 10)( x 1) x 3 3 x 2 10 x We multiply 2 of the parts together first, leaving the third unchanged Now multiply each of the 3 terms in the 1st pair of brackets by each of the 2 terms in the 2nd Polynomials Expanding Cubic Functions e.g. 1 ( x 5)( x 2)( x 1) (x 2 2 x 5 x 10)( x 1) ( x 2 3 x 10)( x 1 ) x 3 3 x 2 10 x x 2 3 x 10 x 3 2 x 2 13 x 10 We multiply 2 of the parts together first, leaving the third unchanged Now multiply each of the 3 terms in the 1st pair of brackets by each of the 2 terms in the 2nd Polynomials Exercise Expand the brackets in the following: ( x 3)( x 1)( x 2) Solution: ( x 3)( x 1)( x 2) ( x 2 x 3 x 3)( x 2) ( x 2 2 x 3)( x 2) x3 2x2 3x 2x2 4x 6 x3 7x 6 Answer: x 7x 6 3 Polynomials Factorising Simple Cubics Some cubic functions which contain a common factor can be factorised by inspection. ( Others are best done using the Factor Theorem which is covered later ). e.g. Factorise fully the following: f ( x) x 3 4 x 2 5 x Solution: Common factor: 2 f ( x ) x ( x 4 x 5) We must now factorise the quadratic. Trinomial factors: f ( x) x( x 5)( x 1) Polynomials Exercise Factorise fully the following cubic: x 2x 8x 3 Solution: 2 x 3 2x 2 8x 2 x ( x 2 x 8) x( x 2)( x 4) Polynomials Summary The degree of a polynomial is the highest x power in the expression. Add or subtract polynomials by column addition or subtraction, or by collecting like terms. Multiply polynomials using any method that helps you to remember to multiply every term in one expression by every term in the other. Solve identities by equating co-efficients. Polynomials Practice Problems Exercise 9 – choose questions to work from in class and at home, but this is a key topic to help you improve your algebra. If this topic is an area of weakness for you, complete all questions. Mark them yourselves, and bring to me any problems that are wrong but you don’t know why.