Use synthetic substitution to find $f(-2)$ and $f(4)$ for each function.

41. $f(x)=x^2-3$
42. $f(x)=x^2-5 x+4$
43. $f(x)=x^3+4 x^2-3 x+2$
44. $f(x)=2 x^4-3 x^3+1

By using synthetic substitution, we have evaluated the given polynomial functions at x = − 2 and x = 4 . The results are: f ( − 2 ) = 1 , 18 , 16 , 57 and f ( 4 ) = 13 , 0 , 118 , 321 for the respective functions. Each evaluation was performed step-by-step using the synthetic substitution method.
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Answered by Anonymous | 2025-07-04

Evaluate each polynomial using synthetic substitution for x = − 2 and x = 4 .
For f ( x ) = x 2 − 3 , find f ( − 2 ) = 1 and f ( 4 ) = 13 .
For f ( x ) = x 2 − 5 x + 4 , find f ( − 2 ) = 18 and f ( 4 ) = 0 .
For f ( x ) = x 3 + 4 x 2 − 3 x + 2 , find f ( − 2 ) = 16 and f ( 4 ) = 118 .
For f ( x ) = 2 x 4 − 3 x 3 + 1 , find f ( − 2 ) = 57 and f ( 4 ) = 321 .
The values are: f ( − 2 ) = 1 , 18 , 16 , 57 and f ( 4 ) = 13 , 0 , 118 , 321 for the given functions, respectively. f ( − 2 ) = 1 , 18 , 16 , 57 ; f ( 4 ) = 13 , 0 , 118 , 321 ​

Explanation

Understanding Synthetic Substitution We are asked to use synthetic substitution to evaluate the given polynomial functions at x = − 2 and x = 4 . Synthetic substitution is a shortcut method for polynomial division that can also be used to evaluate a polynomial at a specific value. The remainder obtained from synthetic division is the value of the polynomial at that specific value.

Evaluating f(x) = x^2 - 3 For f ( x ) = x 2 − 3 :
To find f ( − 2 ) , we perform synthetic substitution with x = − 2 :
-2 | 1 0 -3



-2 4



1 -2 1



So, f ( − 2 ) = 1 .
To find f ( 4 ) , we perform synthetic substitution with x = 4 :
4 | 1 0 -3 | 4 16 |-------------- 1 4 13
So, f ( 4 ) = 13 .

Evaluating f(x) = x^2 - 5x + 4 For f ( x ) = x 2 − 5 x + 4 :
To find f ( − 2 ) , we perform synthetic substitution with x = − 2 :
-2 | 1 -5 4



-2 14



1 -7 18



So, f ( − 2 ) = 18 .
To find f ( 4 ) , we perform synthetic substitution with x = 4 :
4 | 1 -5 4 | 4 -4 |------------- 1 -1 0
So, f ( 4 ) = 0 .

Evaluating f(x) = x^3 + 4x^2 - 3x + 2 For f ( x ) = x 3 + 4 x 2 − 3 x + 2 :
To find f ( − 2 ) , we perform synthetic substitution with x = − 2 :
-2 | 1 4 -3 2



-2 -4 14



1 2 -7 16



So, f ( − 2 ) = 16 .
To find f ( 4 ) , we perform synthetic substitution with x = 4 :
4 | 1 4 -3 2 | 4 32 116 |------------------ 1 8 29 118
So, f ( 4 ) = 118 .

Evaluating f(x) = 2x^4 - 3x^3 + 1 For f ( x ) = 2 x 4 − 3 x 3 + 1 :
To find f ( − 2 ) , we perform synthetic substitution with x = − 2 :
-2 | 2 -3 0 0 1



-4 14 -28 56



2 -7 14 -28 57



So, f ( − 2 ) = 57 .
To find f ( 4 ) , we perform synthetic substitution with x = 4 :
4 | 2 -3 0 0 1 | 8 20 80 320 |------------------ 2 5 20 80 321
So, f ( 4 ) = 321 .

Final Results In summary: For f ( x ) = x 2 − 3 , f ( − 2 ) = 1 and f ( 4 ) = 13 .
For f ( x ) = x 2 − 5 x + 4 , f ( − 2 ) = 18 and f ( 4 ) = 0 .
For f ( x ) = x 3 + 4 x 2 − 3 x + 2 , f ( − 2 ) = 16 and f ( 4 ) = 118 .
For f ( x ) = 2 x 4 − 3 x 3 + 1 , f ( − 2 ) = 57 and f ( 4 ) = 321 .

Examples
Synthetic substitution is a useful tool not just for evaluating polynomials, but also in fields like computer graphics. For instance, when rendering curves and surfaces, evaluating polynomial functions at various points is crucial. By efficiently finding the values of these functions, synthetic substitution helps in creating smooth and accurate visual representations. This method is also applied in engineering to analyze system behaviors modeled by polynomial equations, allowing engineers to predict outcomes and optimize designs.

Answered by GinnyAnswer | 2025-07-04