Expand the product (515 - 0.27x)(368 - 0.27x)(98 - 0.18x) and write it as a cubic in x.

Step-by-step solution

1. 1

   Choose an order that keeps the arithmetic manageable. A triple product is best done as two multiplications, not one. Multiply the two brackets that share the coefficient $-0.27$ first, then distribute the third bracket over the resulting quadratic. Predict the shape too: three linear factors give a cubic, with leading coefficient $(-0.27)(-0.27)(-0.18)$, which is negative.

2. 2

   Multiply the first two brackets with FOIL.

   $515\cdot 368=189520,\qquad -(515+368)(0.27x)=-883\cdot 0.27x=-238.41x,\qquad (0.27x)^2=0.0729x^2$

   so

   $(515-0.27x)(368-0.27x)=0.0729x^2-238.41x+189520$

   Adding the two constants before multiplying by $0.27$ saves a step and a rounding risk.

3. 3

   Distribute the $98$ from the third bracket.

   $98\cdot 0.0729x^2=7.1442x^2,\qquad 98\cdot(-238.41x)=-23364.18x,\qquad 98\cdot 189520=18572960$

4. 4

   Distribute the $-0.18x$ from the third bracket. Each power of $x$ goes up by one:

   $-0.18x\cdot 0.0729x^2=-0.013122x^3,\qquad -0.18x\cdot(-238.41x)=+42.9138x^2,\qquad -0.18x\cdot 189520=-34113.6x$

5. 5

   Collect like terms. The two $x^2$ pieces and the two $x$ pieces combine:

   $x^2:\ 7.1442+42.9138=50.058,\qquad x:\ -23364.18-34113.6=-57477.78$

   giving

   $-0.013122x^3+50.058x^2-57477.78x+18572960$

6. 6

   Check the two easiest coefficients exactly. The constant term must be the product of the constants, $515\cdot 368\cdot 98=18572960$ ✓, and the leading coefficient must be $(-0.27)(-0.27)(-0.18)=-0.013122$ ✓. Both endpoints of the polynomial match, which is the fastest audit of a long expansion.