This presentation is aimed to help students when they have to deal with factoring polynomials

1. FACTORING POLYNOMIALS

14. (x 2 – 3x + 2) : (x – 2) We write this number here 2 1 -3 2

15. (x 2 – 3x + 2) : (x – 2) 2 1 -3 2 and then, we write the coefficients of the polynomial

18. “ An integer root of a polynomial, is always a divisor of its constant term” Example: Which integer numbers could be roots of the polynomial P(x) = - 3x 5 + 4x 2 – 5x -3 ? Answer: The divisors of the constant term They are only +1, -1, +3 y -3

19. It is time to use Ruffini’s rule to make the following divisions faster: P(x):(x-1) P(x):(x+1) P(x):(x-3) and P(x):(x+3) ( x-(-1) ) ( x-(3) )

22. No -6 No +6 No -3 (x-3) Yes +3 No -2 (x-2) Yes +2 (x+1) Yes -1 No +1 Factor Is it a root? Yes/No Possible integer root

25. In a diagram form, we can say: P(x) Possible integer roots: Divisors of the constant term: a, b, c… a is a root b is not a root (x – a) is a factor

26. P(x) Possible integer roots: Divisors of the constant term: a, b, c… a is a root b is not a root (x – a) is a factor If the remainder of the division is zero… P(x) x-a 0 C(x) When the remainder of the division is zero… P(x) x-a 0 C(x)

27. P(x) Possible integer roots: Divisors of the constant term: a, b, c… a is a root b is not a root (x – a) is a factor If the numerical value P(a) = 0 P(a) = 0

28. P(x) Possible integer roots: Divisors of the constant term: a, b, c… a is a root b is not a root (x – a) is a factor If “a” is a solution of the equation P(x) = 0

29. The factors will be: (x-a), (x-b), (x-c), etc… Therefore, if we know every root of the polynomial P(x), so to speak, a, b, c, etc…