x^3+9x^2+6x-16
This solution đơn hàng with finding the roots (zeroes) of polynomials.
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Step by Step Solution
Step 1 :
Equation at the over of step 1 : (((x3) + 32x2) + 6x) - 16Step 2 :
Checking for a perfect cube :2.1x3+9x2+6x-16 is not a perfect cube Trying khổng lồ 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 và 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 mix of methods aimed at finding values ofxfor which F(x)=0 Rational Roots demo 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 và 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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| -1 | 1 | -1.00 | -14.00 | ||||||
| -2 | 1 | -2.00 | 0.00 | x+2 | |||||
| -4 | 1 | -4.00 | 40.00 | ||||||
| -8 | 1 | -8.00 | 0.00 | x+8 | |||||
| -16 | 1 | -16.00 | -1904.00 | ||||||
| 1 | 1 | 1.00 | 0.00 | x-1 | |||||
| 2 | 1 | 2.00 | 40.00 | ||||||
| 4 | 1 | 4.00 | 216.00 | ||||||
| 8 | 1 | 8.00 | 1120.00 | ||||||
| 16 | 1 | 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 chú ý that q and p. Originate from P/Q reduced lớn 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")
| dividend | x3 | + | 9x2 | + | 6x | - | 16 | ||
| -divisor | * x2 | x3 | - | x2 | |||||
| remainder | 10x2 | + | 6x | - | 16 | ||||
| -divisor | * 10x1 | 10x2 | - | 10x | |||||
| remainder | 16x | - | 16 | ||||||
| -divisor | * 16x0 | 16x | - | 16 | |||||
| remainder | 0 |
Quotient : x2+10x+16 Remainder: 0
Trying khổng lồ factor by splitting the middle term2.5Factoring 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 | = | 10 | That"s it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, 2 & 8x2 + 2x+8x + 16Step-4 : địa chỉ up the first 2 terms, pulling out lượt thích factors:x•(x+2) địa chỉ cửa hàng 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