A polynomial function can model the height of a falling object over time. For instance, the function $$h(t) = -16t^2 + 150$$ describes the height in feet of a stone $$t$$ seconds after it is dropped from a cliff that is $$150$$ feet tall. To find the initial height of the object—the height at the exact moment it is dropped—evaluate the function at $$t = 0$$ seconds. Substituting $$0$$ for $$t$$ gives $$h(0) = -16(0)^2 + 150$$. The squared term becomes $$0$$, and multiplying by $$-16$$ still yields $$0$$. Adding $$150$$ results in $$150$$. Thus, the initial height is $$150$$ feet, confirming it matches the height of the cliff.

Using $$-16t^2 + 150$$ to Model the Height of a Dropped Stone

To evaluate the height of a falling object using a polynomial function, substitute the given time into the equation. For example, the function $$h(t) = -16t^2 + 175$$ models the height in feet of a ball $$t$$ seconds after being dropped from a $$175$$-foot bridge. To find the ball's height after $$3$$ seconds, evaluate the function at $$t = 3$$. Substitute $$3$$ for $$t$$ to obtain $$h(3) = -16(3)^2 + 175$$. First, square the $$3$$ to get $$9$$. Next, multiply $$-16$$ by $$9$$ to get $$-144$$. Finally, add $$-144$$ and $$175$$ to arrive at $$31$$. Therefore, the height of the ball after $$3$$ seconds is $$31$$ feet.

Using $$-16t^2 + 175$$ to Model the Height of a Dropped Ball

To apply the process of evaluating a polynomial function to a real-world scenario, consider a ball dropped from a $$250$$-foot building. The height of the ball in feet after $$t$$ seconds is given by the function $$h(t) = -16t^2 + 250$$. To determine the ball's height at $$t = 2$$ seconds, substitute $$2$$ into the function: $$h(2) = -16(2)^2 + 250$$. Simplify the exponent to obtain $$-16(4) + 250$$. Multiplying $$-16$$ by $$4$$ gives $$-64$$, and adding $$250$$ results in $$186$$. Therefore, the height of the ball after $$2$$ seconds is $$186$$ feet.

Using $$-16t^2 + 250$$ to Model the Height of a Dropped Ball

To evaluate $$(f + g)(3)$$ for the functions $$f(x) = 3x^2 - 5x + 7$$ and $$g(x) = x^2 - 4x - 3$$, there are two methods. The first method evaluates the combined function $$(f + g)(x) = 4x^2 - 9x + 4$$ by substituting $$x = 3$$, yielding $$4(3)^2 - 9 \cdot 3 + 4 = 36 - 27 + 4 = 13$$. The second method evaluates $$f(3)$$ and $$g(3)$$ separately and adds the results. First, find $$f(3) = 3(3)^2 - 5(3) + 7 = 27 - 15 + 7 = 19$$. Next, find $$g(3) = (3)^2 - 4(3) - 3 = 9 - 12 - 3 = -6$$. Finally, add the two values: $$(f + g)(3) = f(3) + g(3) = 19 + (-6) = 13$$. Both methods produce the same result.

Evaluating $$(f+g)(3)$$ for $$f(x) = 3x^2 - 5x + 7$$ and $$g(x) = x^2 - 4x - 3$$

To evaluate $$(f - g)(-2)$$ for the functions $$f(x) = 3x^2 - 5x + 7$$ and $$g(x) = x^2 - 4x - 3$$, use the combined difference function $$(f - g)(x) = 2x^2 - x + 10$$. Substitute $$-2$$ for $$x$$ in the expression. Using parentheses for substitution gives $$2(-2)^2 - (-2) + 10$$. Evaluate the exponent first to get $$2(4) - (-2) + 10$$. Perform the multiplication and handle the double negative to obtain $$8 + 2 + 10$$. Add the numbers to find the final value: $$(f - g)(-2) = 20$$.

Evaluating $$(f-g)(-2)$$ for $$f(x) = 3x^2 - 5x + 7$$ and $$g(x) = x^2 - 4x - 3$$

The Remainder Theorem states that if a polynomial function $$f(x)$$ is divided by a binomial of the form $$x - c$$, then the remainder of that division is exactly equal to the value of the function evaluated at $$c$$, which is $$f(c)$$. This theorem can be logically derived by expressing the division in function notation: to get the dividend $$f(x)$$, multiply the quotient $$q(x)$$ by the divisor $$x - c$$, and add the remainder $$r$$. This gives the equation $$f(x) = q(x)(x - c) + r$$. If this equation is evaluated at $$c$$, it yields $$f(c) = q(c)(c - c) + r$$, which simplifies to $$f(c) = q(c)(0) + r$$, leaving $$f(c) = r$$.

Remainder Theorem

A polynomial division problem can be checked by verifying that the dividend equals the product of the quotient and the divisor, plus the remainder. When dividing a polynomial function $$f(x)$$ by a binomial divisor of the form $$x - c$$, this relationship can be written in function notation as $$f(x) = q(x)(x - c) + r$$, where $$q(x)$$ is the quotient and $$r$$ is the remainder. Evaluating this function at $$x = c$$ involves substituting $$c$$ for $$x$$, resulting in the equation $$f(c) = q(c)(c - c) + r$$. Because the term $$(c - c)$$ simplifies to $$0$$, the equation becomes $$f(c) = q(c)(0) + r$$, which reduces to $$f(c) = r$$. This mathematical proof demonstrates that evaluating the original polynomial function at $$c$$ yields the exact value of the remainder.

Polynomial Division Check in Function Notation

To evaluate a polynomial function for a specific input, substitute the given numerical or algebraic value for the variable throughout the polynomial expression. Once the substitution is made, apply the order of operations to simplify the expression and calculate the final result. This procedure builds on the general concept of evaluating a mathematical function, which involves finding the output $$f(x)$$ for a particular input $$x$$.

Claude

A **polynomial function** is a function whose range values are defined by a polynomial expression. For example, the functions $$f(x) = x^2 + 5x + 6$$ and $$g(x) = 3x - 4$$ are considered polynomial functions because their defining expressions, $$x^2 + 5x + 6$$ and $$3x - 4$$, are polynomials.

Polynomial Function

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Evaluating a Polynomial Function

Just as with polynomials, polynomial functions can be combined using addition and subtraction. For any two functions $$f(x)$$ and $$g(x)$$, their sum and difference are defined by the operations:

$$(f + g)(x) = f(x) + g(x)$$

$$(f - g)(x) = f(x) - g(x)$$

To find the sum or difference of two polynomial functions, substitute their defining polynomial expressions into these equations and then combine the like terms.

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