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kati45 [8]
3 years ago
13

Given x^2 -4x+4 is a factor of the polynomial 2x^4 - 11x^3 +68x-80, completely factor the polynomial using algebraic methods.​

Mathematics
1 answer:
asambeis [7]3 years ago
4 0
Answer:

2x^4-11x^3+68x-80

2x^4-4x^3-7x^3+14x^2-14x^2+28x+40x-80

2x^3(x-2)-7x^2(x-2)-14x(x-2)+40(x-2)

(x-2)(2x^3-7x^2-14x+40)

(x-2)(2x^3-4x^2-3x^2+6x-20x+40)

(x-2)(2x^2(x-2)-3x(x-2)-20(x-2))

(x-2)(x-2)(2x^3-3x-20)

(x-2)(x-2)(2x^2+5x-8x-20)

(x-2)(x-2)(x(2x+5)-4(2x+5))

(x-2)(x-2)(2x+5)(x-4)

(x-2)^2(2x+5)(x-4)
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Below

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x=\frac{\text{W}_0\left(\frac{\ln \left(5\right)}{2e^{\frac{3\ln \left(5\right)}{2}}}\right)}{\ln \left(5\right)}+\frac{3}{2}

Step-by-step explanation:

5^{\left(-x\right)}+7=2x+4\\\\\mathrm{Prepare}\:5^{\left(-x\right)}+7=2x+4\:\mathrm{for\:Lambert\:form}:\quad 1=\left(2x-3\right)e^{\ln \left(5\right)x}\\\\\mathrm{Rewrite\:the\:equation\:with\:}\\\left(x-\frac{3}{2}\right)\ln \left(5\right)=u\mathrm{\:and\:}x=\frac{u}{\ln \left(5\right)}+\frac{3}{2}\\\\1=\left(2\left(\frac{u}{\ln \left(5\right)}+\frac{3}{2}\right)-3\right)e^{\ln \left(5\right)\left(\frac{u}{\ln \left(5\right)}+\frac{3}{2}\right)}

Simplify\\\\\mathrm{Rewrite}\:1=\frac{2e^{u+\frac{3}{2}\ln \left(5\right)}u}{\ln \left(5\right)}\:\\\\\mathrm{in\:Lambert\:form}:\quad \frac{e^{\frac{2u+3\ln \left(5\right)}{2}}u}{e^{\frac{3\ln \left(5\right)}{2}}}=\frac{\ln \left(5\right)}{2e^{\frac{3\ln \left(5\right)}{2}}}

\mathrm{Solve\:}\:\frac{e^{\frac{2u+3\ln \left(5\right)}{2}}u}{e^{\frac{3\ln \left(5\right)}{2}}}=\frac{\ln \left(5\right)}{2e^{\frac{3\ln \left(5\right)}{2}}}:\quad u=\text{W}_0\left(\frac{\ln \left(5\right)}{2e^{\frac{3\ln \left(5\right)}{2}}}\right)\\\\\mathrm{Substitute\:back}\:u=\left(x-\frac{3}{2}\right)\ln \left(5\right),\:\mathrm{solve\:for}\:x

\mathrm{Solve\:}\:\left(x-\frac{3}{2}\right)\ln \left(5\right)=\text{W}_0\left(\frac{\ln \left(5\right)}{2e^{\frac{3\ln \left(5\right)}{2}}}\right):\\\quad x=\frac{\text{W}_0\left(\frac{\ln \left(5\right)}{2e^{\frac{3\ln \left(5\right)}{2}}}\right)}{\ln \left(5\right)}+\frac{3}{2}

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Carmen bought a desktop computer and a laptop computer. Before finance charges, the laptop cost $200 less than the desktop. She
stiks02 [169]
Let x = cost of desktop computer; then x - 250 = cost of laptop computer

converting % to decimals, 9% = 0.09 and 6% = 0.06

Then, 0.09x + (x - 250)0.06 = $300
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then x - 250 = $1850

Proof: $2100•0.09 + 1850•0.06 = $189 + $111 = $300
5 0
2 years ago
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