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Mathematics made simple © KS Polynomials A polynomial in x is an expression with positive integer powers of x. Degree of Polynomial Terminology 5x is a term of 5x + 2 and 5 is the coefficient of x. The degree of a polynomial is the highest power of x. A polynomial of degree 1 is called linear. A polynomial of degree 2 is called quadratic. A polynomial of degree 3 is called cubic. A polynomial of degree 4 is called quartic.

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Mathematics made simple © KS Adding and subtracting polynomials Like terms: Like terms have identical letters and powers. Simplify (5x 3 – 4xy + 8) + (2x 3 + 11x + 1)= 7x 3 – 4xy + 11x + 9 (5x 4 – x 3 + 8x) + (2x 4 + 11x + 6)=7x 4 – x 3 + 19x +6 The usual convention is to write the polynomial with the highest power of x first. Simplify (3x 2 + 5x – 11) – (x 2 – 7x – 20) = 2x 2 + 12x + 9 Simplify 3x 2 + 5x – 11 – x 2 +7x + 20 Take care when subtracting

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Mathematics made simple © KS Expanding brackets Simplify 3(2x 2 – 5x + 4) – 2(x 2 – 3x – 1) Expand brackets: 6x 2 – 15x + 12 – 2x 2 + 6x + 2 Collect like terms:= 4x 2 – 9x + 14 f(x) = 4x 3 – 2x 2 + 5x - 11 g(x) = x 3 + 3x 2 - 4x + 5 Simplify f(x) + 2g(x) f(x)+ 2g(x)= 4x 3 – 2x 2 + 5x – 11 + 2(x 3 + 3x 2 – 4x + 5) = 4x 3 – 2x 2 + 5x – 11 + 2x 3 + 6x 2 – 8x + 10 = 6x 3 + 4x 2 - 3x – 1

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Mathematics made simple © KS Expanding two or more brackets (a + b)(c + d) =ac + ad + bc + bd (x + 4)(x – 6)= x 2 – 6x + 4x - 24 = x 2 – 2x - 24 Collect like terms (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 - 2ab + b 2 (a - b)(a + b)= a 2 - b 2 (a – b) 3 = a 3 - 3a 2 b + 3ab 2 + b 3 Expand two brackets then multiply by the third.

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Mathematics made simple © KS Three brackets Expand two of the brackets, then multiply by the third bracket Expand and simplify the polynomial (2x + 1)(x – 3)(x + 4).

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Mathematics made simple © KS Equate coefficients of corresponding powers of x Write down the values of a, b and c Find the values of a, b and c: Expand the left hand side a = 3, b = –1, c = 5 Identical polynomials ( )

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Mathematics made simple © KS Substitute x = 2 into the polynomial and equate it to 10. Given that P(2) = 10, find the value of k. Polynomial substitution

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