1. Let f(x) = x^3 + 3x ^2 + 4x - 7 and g(x) = -7x^4 + 5x^3 +x^2 - 7 . What is the coefficient of x^3 in the sum f(x) + g(x)?
2. Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1. Let a be a constant. What is the largest possible degree of f(x)+a*g(x)?
3. Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1. Let b be a constant. What is the smallest possible degree of the polynomial f(x) + b* g(x)?
4. Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1. What is the degree of f(x)*g(x)?
5. The degree of the polynomial p(x) is 11, and the degree of the polynomial q(x) is 7. Find all possible degrees of the polynomial p(x)+q(x).
6. Find t if the expansion of the product of x^3 - 4x^2 + 2x - 5 and x^2 + tx - 7 has no x^2 term.
1. You just need to add up the x^3 terms, which is just x^3+5x^3 = 6x^3, which means the answer is 6.
2. Since both of the polynomials are of degree 4, the largest possible value of the degree of f(x)+a*g(x) is still 4, since multiplying a polynomial by a constant doesn't change its degree.
3. To find the smallest possible degree, try to cancel out the x^4 term. To do that, set b = -1/2. That will make the x^4 term of g(x) equal to -1, which cancels out with the x^4 term of f(x), but notice that the x^2 terms also cancel out. That means the smallest possible degree is 1.
4. since both of the polynomials are of degree 4, the degree of f(x)*g(x) is just 4+4=8.
5. very similar to question 2, the answer is 11.
6. hint: this is the same question as solving this equation: