Final answer:
To express sinA sin2A sin3A sin4A as a polynomial in x where sin²A=x, use trigonometric identities and angle sum formulas to rewrite each term in terms of sinA and cosA, then substitute sin²A with x and cos²A with 1-x to find the polynomial with its coefficients.
Step-by-step explanation:
If sin²A=x, we want to express sinA sin2A sin3A sin4A as a polynomial in x. First, notice that sin2A=2sinAcosA and we can use the identity cos²A=1-sin²A to express cosA in terms of x.
Similarly, we can express sin3A and sin4A using angle sum and double angle formulas, such as sin(2A+A)=sin2AcosA+cos2AsinA and sin2(2A)=2sin2Acos2A. After expressing all terms in terms of sinA and cosA, we will replace sin²A with x and cos²A with 1-x.
Combining these expressions yields a polynomial in x, and the sum of the coefficients is obtained by substituting x=1 into the polynomial (since every term's coefficient contributes to the total when x=1). This process will reveal that the sum of the coefficients of the polynomial is either 0, 40, 168, or 336. Without fully expanding the polynomial, the exact option cannot be determined in the scope of this answer.