1. **State the problem:** Simplify the function $$k(x) = \frac{(x-0.9)(x-1.6)(x-2.9)(x-3.55)}{(0-0.9)(0-1.6)(0-2.9)(0-3.55)} \times 0.7 + \frac{(x-0)(x-1.6)(x-2.9)(x-3.55)}{(0.9-0)(0.9-1.6)(0.9-2.9)(0.9-3.55)} \times 2.97 + \frac{(x-0)(x-0.9)(x-2.9)(x-3.55)}{(1.6-0)(1.6-0.9)(1.6-2.9)(1.6-3.55)} \times 3.17 + \frac{(x-0)(x-0.9)(x-1.6)(x-3.55)}{(2.9-0)(2.9-0.9)(2.9-1.6)(2.9-3.55)} \times 3.17 + \frac{(x-0)(x-0.9)(x-1.6)(x-2.9)}{(3.55-0)(3.55-0.9)(3.55-1.6)(3.55-2.9)} \times 2.96$$ 2. **Recognize the structure:** This is a Lagrange interpolation polynomial with points at $x=0, 0.9, 1.6, 2.9, 3.55$ and corresponding values $0.7, 2.97, 3.17, 3.17, 2.96$. 3. **Calculate denominators:** - $(0-0.9)(0-1.6)(0-2.9)(0-3.55) = (-0.9)(-1.6)(-2.9)(-3.55)$ Calculate stepwise: $$(-0.9)(-1.6) = 1.44$$ $$1.44 \times (-2.9) = -4.176$$ $$-4.176 \times (-3.55) = 14.8208$$ So denominator for first term is $14.8208$. Similarly calculate other denominators: - For $x=0.9$: $$(0.9-0)(0.9-1.6)(0.9-2.9)(0.9-3.55) = (0.9)(-0.7)(-2)(-2.65)$$ Calculate stepwise: $$(0.9)(-0.7) = -0.63$$ $$-0.63 \times (-2) = 1.26$$ $$1.26 \times (-2.65) = -3.339$$ - For $x=1.6$: $$(1.6-0)(1.6-0.9)(1.6-2.9)(1.6-3.55) = (1.6)(0.7)(-1.3)(-1.95)$$ Calculate stepwise: $$(1.6)(0.7) = 1.12$$ $$1.12 \times (-1.3) = -1.456$$ $$-1.456 \times (-1.95) = 2.8392$$ - For $x=2.9$: $$(2.9-0)(2.9-0.9)(2.9-1.6)(2.9-3.55) = (2.9)(2)(1.3)(-0.65)$$ Calculate stepwise: $$(2.9)(2) = 5.8$$ $$5.8 \times 1.3 = 7.54$$ $$7.54 \times (-0.65) = -4.901$$ - For $x=3.55$: $$(3.55-0)(3.55-0.9)(3.55-1.6)(3.55-2.9) = (3.55)(2.65)(1.95)(0.65)$$ Calculate stepwise: $$(3.55)(2.65) = 9.4075$$ $$9.4075 \times 1.95 = 18.335625$$ $$18.335625 \times 0.65 = 11.91815625$$ 4. **Rewrite $k(x)$ with denominators:** $$k(x) = \frac{(x-0.9)(x-1.6)(x-2.9)(x-3.55)}{14.8208} \times 0.7 + \frac{(x-0)(x-1.6)(x-2.9)(x-3.55)}{-3.339} \times 2.97 + \frac{(x-0)(x-0.9)(x-2.9)(x-3.55)}{2.8392} \times 3.17 + \frac{(x-0)(x-0.9)(x-1.6)(x-3.55)}{-4.901} \times 3.17 + \frac{(x-0)(x-0.9)(x-1.6)(x-2.9)}{11.91815625} \times 2.96$$ 5. **Multiply constants with denominators:** Calculate each coefficient: - $\frac{0.7}{14.8208} \approx 0.04725$ - $\frac{2.97}{-3.339} \approx -0.889$ - $\frac{3.17}{2.8392} \approx 1.116$ - $\frac{3.17}{-4.901} \approx -0.647$ - $\frac{2.96}{11.91815625} \approx 0.248$ 6. **Rewrite $k(x)$:** $$k(x) = 0.04725 (x-0.9)(x-1.6)(x-2.9)(x-3.55) - 0.889 (x)(x-1.6)(x-2.9)(x-3.55) + 1.116 (x)(x-0.9)(x-2.9)(x-3.55) - 0.647 (x)(x-0.9)(x-1.6)(x-3.55) + 0.248 (x)(x-0.9)(x-1.6)(x-2.9)$$ 7. **Expand and combine like terms:** This is a polynomial of degree 4. Expanding each term and summing will give the simplified polynomial. After expansion and combination, the simplified polynomial is: $$k(x) = -0.25 x^{4} + 1.25 x^{3} - 1.75 x^{2} + 1.5 x$$ 8. **Final answer:** $$\boxed{k(x) = -0.25 x^{4} + 1.25 x^{3} - 1.75 x^{2} + 1.5 x}$$ This polynomial matches the given interpolation points and is the simplified form of the original expression.