Find all the zeros of the polynomial function Use the Ration

Find all the zeros of the polynomial function. Use the Rational Zero Theorem, Descartes\'s Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero. f(x) = x^3 - 5x^2 - 17x + 21 The zeros of the function are .

Solution

Given, f(x)=x³-5x²-17x+21

By factorising. f(x)= (x + 3) • (x - 1) • (x - 7)

Step by step solution :

Step 1 :Equation at the end of step 1 :

Step 2 :Checking for a perfect cube :

2.1    x³-5x²-17x+21 is not a perfect cube

Trying to factor by pulling out :

2.2      Factoring: x³-5x²-17x+21

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: -17x+21
Group 2: x³-5x²

Pull out from each group separately :

Group 1: (-17x+21) • (1) = (17x-21) • (-1)
Group 2: (x-5) • (x²)

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

2.3    Find roots (zeroes) of :       F(x) = x³-5x²-17x+21
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is 21.

The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,7 ,21
Let us test ....

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
   x³-5x²-17x+21
can be divided by 3 different polynomials,including by x-7

Polynomial Long Division :

2.4    Polynomial Long Division
Dividing : x³-5x²-17x+21   .....    (\"Dividend\")
By         :    x-7 .......... (\"Divisor\")

Quotient : x²+2x-3 Remainder: 0

Trying to factor by splitting the middle term

2.5     Factoring x²+2x-3

The first term is, x² its coefficient is 1 .
The middle term is, +2x its coefficient is 2 .
The last term, \"the constant\", is -3

Step-1 : Multiply the coefficient of the first term by the constant 1 • -3 = -3

Step-2 : Find two factors of -3 whose sum equals the coefficient of the middle term, which is   2 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 3
                     x² - 1x + 3x - 3

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-1)
              Add up the last 2 terms, pulling out common factors :
                    3 • (x-1)
Step-5 : Add up the four terms of step 4 :
                    (x+3)  •  (x-1)
             Which is the desired factorization

Final result :

ie, x=-3, 1, 7

	P		Q		P/Q		F(P/Q)		Divisor
	-1		1		-1.00		32.00
	-3		1		-3.00		0.00		x+3
	-7		1		-7.00		-448.00
	-21		1		-21.00		-11088.00
	1		1		1.00		0.00		x-1
	3		1		3.00		-48.00
	7		1		7.00		0.00		x-7
	21		1		21.00		6720.00