Final answer:
The degree of the polynomial y=(10x+1)^2(7x-3)(x+4)^3 is 6, and the leading coefficient is 700, determined by multiplying the highest degree coefficients of each term.
Step-by-step explanation:
To find the degree of the given polynomial function y=(10x+1)^2(7x-3)(x+4)^3, we count the total number of x's when the expression is fully expanded. Each term raises x to a certain power, and these powers add up when terms are multiplied. We have:
- (10x+1)^2: a second-degree term (2 from the exponent)
- (7x-3): a first-degree term (1 from the exponent, which is implied)
- (x+4)^3: a third-degree term (3 from the exponent)
Add the exponents from each term: 2 + 1 + 3 = 6. So, the degree of the given polynomial is 6.
To find the leading coefficient, we look at the coefficients of the highest degree terms when multiplied out. We multiply the highest coefficients of each term: 10^2 * 7 * 1^3 = 100 * 7 * 1. Thus, the leading coefficient is 700.