x/2x^2+(5x-8)^2-176
This solution giao dịch with finding the roots (zeroes) of polynomials.
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Step by Step Solution
Step 1 :
x Simplify — 2Equation at the end of step 1 : x ((— • x2) + (5x - 8)2) - 176 2Step 2 :
Multiplying exponential expressions :2.1 x1 multiplied by x2 = x(1 + 2) = x3Equation at the end of step 2 :x3 (—— + (5x - 8)2) - 176 2
Step 3 :
Rewriting the whole as an Equivalent Fraction :3.1Adding a whole khổng lồ a fraction Rewrite the whole as a fraction using 2 as the denominator :(5x - 8)2 (5x - 8)2 • 2 (5x - 8)2 = ————————— = ————————————— 1 2 Equivalent fraction : The fraction thus generated looks different but has the same value as the whole Common denominator : The equivalent fraction and the other fraction involved in the calculation nói qua the same denominator
Adding fractions that have a common denominator :3.2 Adding up the two equivalent fractions showroom the two equivalent fractions which now have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce khổng lồ lowest terms if possible:
x3 + (5x-8)2 • 2 x3 + 50x2 - 160x + 128 ———————————————— = —————————————————————— 2 2 Equation at the kết thúc of step 3 : (x3 + 50x2 - 160x + 128) ———————————————————————— - 176 2
Step 4 :
Rewriting the whole as an Equivalent Fraction :4.1Subtracting a whole from a fraction Rewrite the whole as a fraction using 2 as the denominator :176 176 • 2 176 = ——— = ——————— 1 2 Checking for a perfect cube :4.2x3 + 50x2 - 160x + 128 is not a perfect cube
Trying lớn factor by pulling out :4.3 Factoring: x3 + 50x2 - 160x + 128 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: x3 + 50x2Group 2: -160x + 128Pull out from each group separately :Group 1: (x + 50) • (x2)Group 2: (5x - 4) • (-32)Bad news !! Factoring by pulling out fails : The groups have no common factor và can not be added up to size a multiplication.
Polynomial Roots Calculator :
4.4 Find roots (zeroes) of : F(x) = x3 + 50x2 - 160x + 128Polynomial Roots Calculator is a phối of methods aimed at finding values ofxfor which F(x)=0 Rational Roots kiểm tra 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 & the Trailing Constant is 128. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,2 ,4 ,8 ,16 ,32 ,64 ,128 Let us demo ....
| -1 | 1 | -1.00 | 337.00 | ||||||
| -2 | 1 | -2.00 | 640.00 | ||||||
| -4 | 1 | -4.00 | 1504.00 | ||||||
| -8 | 1 | -8.00 | 4096.00 | ||||||
| -16 | 1 | -16.00 | 11392.00 | ||||||
| -32 | 1 | -32.00 | 23680.00 | ||||||
| -64 | 1 | -64.00 | -46976.00 | ||||||
| -128 | 1 | -128.00 | -1257344.00 | ||||||
| 1 | 1 | 1.00 | 19.00 | ||||||
| 2 | 1 | 2.00 | 16.00 | ||||||
| 4 | 1 | 4.00 | 352.00 | ||||||
| 8 | 1 | 8.00 | 2560.00 | ||||||
| 16 | 1 | 16.00 | 14464.00 | ||||||
| 32 | 1 | 32.00 | 78976.00 | ||||||
| 64 | 1 | 64.00 | 456832.00 | ||||||
| 128 | 1 | 128.00 | 2896000.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :4.5 Adding up the two equivalent fractions
(x3+50x2-160x+128) - (176 • 2) x3 + 50x2 - 160x - 224 —————————————————————————————— = —————————————————————— 2 2 Checking for a perfect cube :4.6x3 + 50x2 - 160x - 224 is not a perfect cube
Trying to factor by pulling out :4.7 Factoring: x3 + 50x2 - 160x - 224 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: x3 + 50x2Group 2: -160x - 224Pull out from each group separately :Group 1: (x + 50) • (x2)Group 2: (5x + 7) • (-32)Bad news !! Factoring by pulling out fails : The groups have no common factor and can not be added up to khung a multiplication.
Polynomial Roots Calculator :
4.8 Find roots (zeroes) of : F(x) = x3 + 50x2 - 160x - 224See theory in step 4.4 In this case, the Leading Coefficient is 1 và the Trailing Constant is -224. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,2 ,4 ,7 ,8 ,14 ,16 ,28 ,32 ,56 , etc Let us thử nghiệm ....
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| -1 | 1 | -1.00 | -15.00 | ||||||
| -2 | 1 | -2.00 | 288.00 | ||||||
| -4 | 1 | -4.00 | 1152.00 | ||||||
| -7 | 1 | -7.00 | 3003.00 | ||||||
| -8 | 1 | -8.00 | 3744.00 | ||||||
| -14 | 1 | -14.00 | 9072.00 | ||||||
| -16 | 1 | -16.00 | 11040.00 | ||||||
| -28 | 1 | -28.00 | 21504.00 | ||||||
| -32 | 1 | -32.00 | 23328.00 | ||||||
| -56 | 1 | -56.00 | -10080.00 | ||||||
| 1 | 1 | 1.00 | -333.00 | ||||||
| 2 | 1 | 2.00 | -336.00 | ||||||
| 4 | 1 | 4.00 | 0.00 | x - 4 | |||||
| 7 | 1 | 7.00 | 1449.00 | ||||||
| 8 | 1 | 8.00 | 2208.00 | ||||||
| 14 | 1 | 14.00 | 10080.00 | ||||||
| 16 | 1 | 16.00 | 14112.00 | ||||||
| 28 | 1 | 28.00 | 56448.00 | ||||||
| 32 | 1 | 32.00 | 78624.00 | ||||||
| 56 | 1 | 56.00 | 323232.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p lưu ý that q and p. Originate from P/Q reduced khổng lồ its lowest terms In our case this means that x3 + 50x2 - 160x - 224can be divided with x - 4
Polynomial Long Division :
4.9 Polynomial Long Division Dividing : x3 + 50x2 - 160x - 224("Dividend") By:x - 4("Divisor")| dividend | x3 | + | 50x2 | - | 160x | - | 224 | ||
| -divisor | * x2 | x3 | - | 4x2 | |||||
| remainder | 54x2 | - | 160x | - | 224 | ||||
| -divisor | * 54x1 | 54x2 | - | 216x | |||||
| remainder | 56x | - | 224 | ||||||
| -divisor | * 56x0 | 56x | - | 224 | |||||
| remainder | 0 |
Quotient : x2+54x+56 Remainder: 0
Trying lớn factor by splitting the middle term4.10Factoring x2+54x+56 The first term is, x2 its coefficient is 1.The middle term is, +54x its coefficient is 54.The last term, "the constant", is +56Step-1 : Multiply the coefficient of the first term by the constant 1•56=56Step-2 : Find two factors of 56 whose sum equals the coefficient of the middle term, which is 54.
| -56 | + | -1 | = | -57 | ||
| -28 | + | -2 | = | -30 | ||
| -14 | + | -4 | = | -18 | ||
| -8 | + | -7 | = | -15 | ||
| -7 | + | -8 | = | -15 | ||
| -4 | + | -14 | = | -18 | ||
| -2 | + | -28 | = | -30 | ||
| -1 | + | -56 | = | -57 | ||
| 1 | + | 56 | = | 57 | ||
| 2 | + | 28 | = | 30 | ||
| 4 | + | 14 | = | 18 | ||
| 7 | + | 8 | = | 15 | ||
| 8 | + | 7 | = | 15 | ||
| 14 | + | 4 | = | 18 | ||
| 28 | + | 2 | = | 30 | ||
| 56 | + | 1 | = | 57 |
Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored