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Polynomial Arithmetic

From profit/cost models to the flight of a baseball to approximating the motion of a wave, polynomials are a mathematical way to represent values which are sums of powers of variables.

Combining Like Terms

         

What is the coefficient of \( x^ {4} \) in the following polynomial:

\[ -6 x^ {4} + 4 x^3 + 15 x^{4} + 4 x^{15} ? \]

After combining like terms, what is the sum of the coefficients of the \( xy \) and \( x \) terms in the polynomial \[3x+7xy+25y+11x-3xy?\]

After combining like terms, what is the sum of all coefficients of the polynomial \[2x-(x-12y-(6x-5y))?\]

For example, the polynomial \( 2x + 3y + 2 \) has coefficients of \(2, 3 \) and \(2\), and the sum of all coefficients is \( 2 + 3 + 2 = 7 \).

Rewrite the following expression in the form \( ax + by \), and determine the sum of \(a\) and \(b\): \[5x-(4x+3y-(2x-y))+63y.\]

Given \( f(x) = 8x + 5 \) and \( g(x) = 5x + 12 \), what is the value of the constant term in \( f(x) + g(x) \)?

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