x^3+9x^2+6x-16

This solution deals with finding the roots (zeroes) of polynomials.

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Step by Step Solution

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Step 1 :

Equation at the end of step 1 : (((x3) + 32x2) + 6x) - 16

Step 2 :

Checking for a perfect cube :2.1x3+9x2+6x-16 is not a perfect cube

Trying to factor by pulling out :

2.2 Factoring: x3+9x2+6x-16 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: 6x-16Group 2: 9x2+x3Pull out from each group separately :Group 1: (3x-8) • (2)Group 2: (x+9) • (x2)Bad news !! Factoring by pulling out fails : 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+9x2+6x-16Polynomial Roots Calculator is a set of methods aimed at finding values ofxfor 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 integersThe Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 and the Trailing Constant is -16. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,2 ,4 ,8 ,16 Let us test ....

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PQP/QF(P/Q)Divisor
-11 -1.00 -14.00
-21 -2.00 0.00x+2
-41 -4.00 40.00
-81 -8.00 0.00x+8
-161-16.00-1904.00
11 1.00 0.00x-1
21 2.00 40.00
41 4.00 216.00
81 8.00 1120.00
161 16.00 6480.00

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+9x2+6x-16can be divided by 3 different polynomials,including by x-1

Polynomial Long Division :

2.4 Polynomial Long Division Dividing : x3+9x2+6x-16("Dividend") By:x-1("Divisor")

dividendx3+9x2+6x-16
-divisor* x2x3-x2
remainder10x2+6x-16
-divisor* 10x110x2-10x
remainder16x-16
-divisor* 16x016x-16
remainder0

Quotient : x2+10x+16 Remainder: 0

Trying to factor by splitting the middle term

2.5Factoring x2+10x+16 The first term is, x2 its coefficient is 1.The middle term is, +10x its coefficient is 10.The last term, "the constant", is +16Step-1 : Multiply the coefficient of the first term by the constant 1•16=16Step-2 : Find two factors of 16 whose sum equals the coefficient of the middle term, which is 10.

-16+-1=-17
-8+-2=-10
-4+-4=-8
-2+-8=-10
-1+-16=-17
1+16=17
2+8=10That"s it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, 2 and 8x2 + 2x+8x + 16Step-4 : Add up the first 2 terms, pulling out like factors:x•(x+2) Add up the last 2 terms, pulling out common factors:8•(x+2) Step-5:Add up the four terms of step4:(x+8)•(x+2)Which is the desired factorization