Question

1. given $f(x)=x^3 + 3x^2 - 5$

evaluate when $x = -7$

1. given

$f(x)=2x^2 + 3x - 5$
evaluate when $x = 4$

1. if $f(x)=2x^4 + 3x^2 - 3$,

then what is the remainder
when $f(x)$ is divided by
$x - 1$

1. if $f(x)=2x^2 - 5x + 5$,

then what is the remainder
when $f(x)$ is divided by
$x - 9$

1. if $f(x)=x^4 - x^3 - 3$

find $f(-4)$

1. if

$f(x)=3x^4 - 2x^3 - x^2 + 4x - 3$
find $f(-6)$

1. given

$f(x)= -3x^3 + 4x^2 - 10x - 3$
evaluate when $x = 2$

1. if $f(x)=5x^3 + x - 2$,

then what is the remainder
when $f(x)$ is divided by
$x + 1$
solutions

1. -201
2. 39
3. 2 (you have to use a +1)
4. 122 (you have to use +9)
5. 317
6. 4,257
7. -31
8. -8

Explanation:

Step1: Identify the problem type

These are problems involving evaluating polynomial functions at specific values or finding remainders when dividing polynomials (which can be solved using the Remainder Theorem). Let's take problem 1 as an example: \( f(x)=x^{3}+3x^{2}-5 \), evaluate at \( x = - 7 \).

Step2: Substitute \( x=-7 \) into the function

First, calculate \( x^{3} \) when \( x=-7 \): \( (-7)^{3}=-343 \). Then, calculate \( 3x^{2} \) when \( x = - 7 \): \( 3\times(-7)^{2}=3\times49 = 147 \). Now, substitute these into the function: \( f(-7)=(-7)^{3}+3\times(-7)^{2}-5=-343 + 147-5 \).

Step3: Perform the arithmetic operations

\( -343+147=-196 \), then \( -196 - 5=-201 \).

Answer:

For problem 1, the answer is \(-201\) (as per the given solutions, and following the same method for other problems: substitute the value of \( x \) into the polynomial and simplify, or use the Remainder Theorem which states that the remainder of dividing \( f(x) \) by \( x - a \) is \( f(a) \)). For example, for problem 3, using Remainder Theorem, divide by \( x - 1 \), so remainder is \( f(1)=2(1)^{4}+3(1)^{2}-3=2 + 3-3 = 2 \), which matches the given solution. Each problem follows similar substitution or Remainder Theorem application.