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cubic polynomials

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How do I use synthetic division to factor cubic polynomials like this

x^3 - 5x^2 - 2x + 24
asked Nov 19, 2013 in ALGEBRA 2 by dkinz Apprentice

2 Answers

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Given polynomial x^3-5x^2-2x+24

By synthetic division

image

x^3-5x^2-2x+24 = (x+2)(x^2-7x+12)

x^2-4x-3x+12

x(x-4)-3(x-4)

(x-4)(x-3)

Factoring of given polynomial is (x+2)(x-4)(x-3)

answered Jan 6, 2014 by ashokavf Scholar
0 votes

The polynomial function x3- 5x2- 2x + 24

From factor theorem when f(c) = 0 then x - c is factor of polynomial.

Identify Rational Zeros  

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

x3- 5x2- 2x + 24 = 0

If p/q is a rational zero, then p is a factor of 24 and q is a factor of 1.

The possible values of p are   ± 1, ±2,  ± 3, ±4, ±6, ±8, ±12 and   ±24.

The possible values for q are ± 1.

So, p/q =   ± 1, ±2,  ± 3, ±4, ±6, ±8, ±12 and   ±24.

Make a table for the synthetic division and test possible  zeros.

p/q 1 -5 -2 24
1 1 -4 -6 30
-1 1 -6 4 20
-2 1 -7 12 0

Since f(-2)  =  0,  x   =  -2 is a zero. The depressed polynomial is   x2 - 7x + 12 = 0

Since the depressed polynomial of this zero,  x2 - 7x + 12, is quadratic, use the  Factorization find the roots of the related quadratic equation

 x2 - 7x + 12 = 0

 x2- 4x - 3x + 12 = 0

x(x - 4) - 3(x - 4) = 0

(x - 4) (x - 3) = 0

x - 4 = 0 and x - 3 = 0

x = 4 and x = 3

Zeros of the cubic polynomial are x = 3, 4 and -2.

Factoring of x3- 5x2- 2x + 24 is (x - 4)(x - 3)(x + 2).

answered Aug 5, 2014 by david Expert

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