### x/2x^2+(5x-8)^2-176

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

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## Step 1 :

x Simplify — 2Equation at the end of step 1 : x ((— • x2) + (5x - 8)2) - 176 2

## Step 2 :

Multiplying exponential expressions :2.1 x1 multiplied by x2 = x(1 + 2) = x3

Equation 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 ....

PQP/QF(P/Q)Divisor
-11 -1.00 337.00
-21 -2.00 640.00
-41 -4.00 1504.00
-81 -8.00 4096.00
-161-16.0011392.00
-321-32.0023680.00
-641-64.00-46976.00
-1281-128.00-1257344.00
11 1.00 19.00
21 2.00 16.00
41 4.00 352.00
81 8.00 2560.00
161 16.0014464.00
321 32.0078976.00
641 64.00456832.00
1281128.002896000.00

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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PQP/QF(P/Q)Divisor
-11 -1.00 -15.00
-21 -2.00 288.00
-41 -4.00 1152.00
-71 -7.00 3003.00
-81 -8.00 3744.00
-141-14.00 9072.00
-161-16.0011040.00
-281-28.0021504.00
-321-32.0023328.00
-561-56.00-10080.00
11 1.00 -333.00
21 2.00 -336.00
41 4.00 0.00x - 4
71 7.00 1449.00
81 8.00 2208.00
141 14.0010080.00
161 16.0014112.00
281 28.0056448.00
321 32.0078624.00
561 56.00323232.00

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 term

4.10Factoring 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