Evaluate the polynomial y= x^3- 5x^2+ 6x + 0.55 at x=1.37

/ Exams & Certification / By

Evaluate the polynomial y= x^3- 5x^2+ 6x + 0.55 at x=1.37 Use 3-digit arithmetic with rounding. Evaluate the percent relative error. (b) Repeat (a) but express y asy= ((x- 5)x +6)x + 0.55

The Correct Answer and Explanation is:

Let’s evaluate the polynomial y=x3−5×2+6x+0.55y = x^3 – 5x^2 + 6x + 0.55y=x3−5×2+6x+0.55 at x=1.37x = 1.37x=1.37 using 3-digit arithmetic and rounding. Then we’ll calculate the percent relative error. For part (b), we’ll repeat the process using the factored form y=((x−5)x+6)x+0.55y = ((x – 5)x + 6)x + 0.55y=((x−5)x+6)x+0.55.

Part (a) – Polynomial Evaluation

We are given the polynomial:y=x3−5×2+6x+0.55y = x^3 – 5x^2 + 6x + 0.55y=x3−5×2+6x+0.55

Substitute x=1.37x = 1.37x=1.37 into the polynomial:y=(1.37)3−5(1.37)2+6(1.37)+0.55y = (1.37)^3 – 5(1.37)^2 + 6(1.37) + 0.55y=(1.37)3−5(1.37)2+6(1.37)+0.55

  1. (1.37)3=2.599073(1.37)^3 = 2.599073(1.37)3=2.599073
  2. (1.37)2=1.8769(1.37)^2 = 1.8769(1.37)2=1.8769, so 5(1.37)2=9.38455(1.37)^2 = 9.38455(1.37)2=9.3845
  3. 6(1.37)=8.226(1.37) = 8.226(1.37)=8.22

Now substitute these into the polynomial:y=2.599073−9.3845+8.22+0.55y = 2.599073 – 9.3845 + 8.22 + 0.55y=2.599073−9.3845+8.22+0.55y=2.599073+8.22−9.3845+0.55y = 2.599073 + 8.22 – 9.3845 + 0.55y=2.599073+8.22−9.3845+0.55y=10.819073−9.3845+0.55=2.984573y = 10.819073 – 9.3845 + 0.55 = 2.984573y=10.819073−9.3845+0.55=2.984573

Rounding to three significant digits:y≈2.98y \approx 2.98y≈2.98

Percent Relative Error

The exact value is calculated using a more precise value for the power terms:

  1. (1.37)3=2.599073(1.37)^3 = 2.599073(1.37)3=2.599073
  2. 5(1.37)2=9.38455(1.37)^2 = 9.38455(1.37)2=9.3845
  3. 6(1.37)=8.226(1.37) = 8.226(1.37)=8.22

For the exact value, the polynomial sum is 2.599073−9.3845+8.22+0.55≈2.9845732.599073 – 9.3845 + 8.22 + 0.55 \approx 2.9845732.599073−9.3845+8.22+0.55≈2.984573. The value rounded to three digits is 2.98.

The percent relative error is given by the formula:Percent Relative Error=∣Exact Value−Approximate ValueExact Value∣×100\text{Percent Relative Error} = \left| \frac{\text{Exact Value} – \text{Approximate Value}}{\text{Exact Value}} \right| \times 100Percent Relative Error=​Exact ValueExact Value−Approximate Value​​×100Percent Relative Error=∣2.984573−2.982.984573∣×100\text{Percent Relative Error} = \left| \frac{2.984573 – 2.98}{2.984573} \right| \times 100Percent Relative Error=​2.9845732.984573−2.98​​×100Percent Relative Error≈∣0.0045732.984573∣×100≈0.153%\text{Percent Relative Error} \approx \left| \frac{0.004573}{2.984573} \right| \times 100 \approx 0.153\%Percent Relative Error≈​2.9845730.004573​​×100≈0.153%

Part (b) – Using the Factored Form

Now, we express the polynomial in factored form:y=((x−5)x+6)x+0.55y = ((x – 5)x + 6)x + 0.55y=((x−5)x+6)x+0.55

Substitute x=1.37x = 1.37x=1.37 into this form:y=((1.37−5)×1.37+6)×1.37+0.55y = ((1.37 – 5) \times 1.37 + 6) \times 1.37 + 0.55y=((1.37−5)×1.37+6)×1.37+0.55

First, calculate the inner expressions:1.37−5=−3.631.37 – 5 = -3.631.37−5=−3.63(−3.63)×1.37=−4.9781(-3.63) \times 1.37 = -4.9781(−3.63)×1.37=−4.9781−4.9781+6=1.0219-4.9781 + 6 = 1.0219−4.9781+6=1.02191.0219×1.37=1.3988031.0219 \times 1.37 = 1.3988031.0219×1.37=1.3988031.398803+0.55=1.9488031.398803 + 0.55 = 1.9488031.398803+0.55=1.948803

Rounding to three significant digits:y≈1.95y \approx 1.95y≈1.95

Percent Relative Error for Factored Form

Now calculate the percent relative error:Percent Relative Error=∣Exact Value−Approximate ValueExact Value∣×100\text{Percent Relative Error} = \left| \frac{\text{Exact Value} – \text{Approximate Value}}{\text{Exact Value}} \right| \times 100Percent Relative Error=​Exact ValueExact Value−Approximate Value​​×100

The exact value is 2.984573, and the approximate value is 1.95:Percent Relative Error=∣2.984573−1.952.984573∣×100\text{Percent Relative Error} = \left| \frac{2.984573 – 1.95}{2.984573} \right| \times 100Percent Relative Error=​2.9845732.984573−1.95​​×100Percent Relative Error=∣1.0345732.984573∣×100≈34.7%\text{Percent Relative Error} = \left| \frac{1.034573}{2.984573} \right| \times 100 \approx 34.7\%Percent Relative Error=​2.9845731.034573​​×100≈34.7%

Conclusion

  • In part (a), the evaluation of the polynomial gives y≈2.98y \approx 2.98y≈2.98, with a percent relative error of about 0.153%0.153\%0.153%.
  • In part (b), using the factored form gives y≈1.95y \approx 1.95y≈1.95, with a percent relative error of about 34.7%34.7\%34.7%.

This significant difference in errors between the two forms highlights how the polynomial’s structure can affect the accuracy of numerical evaluations. The original polynomial gives a much more accurate result than the factored form in this case.

Related Posts

Scroll to Top