Answer :
To determine if [tex]\( x - 5 \)[/tex] is a factor of the polynomial [tex]\( f(x) = 4x^4 - 23x^3 + 6x^2 + 43x + 10 \)[/tex], we can use the Factor Theorem. According to the Factor Theorem, [tex]\( x - a \)[/tex] is a factor of a polynomial [tex]\( f(x) \)[/tex] if and only if [tex]\( f(a) = 0 \)[/tex].
In this case, we need to check if [tex]\( f(5) = 0 \)[/tex].
Let's evaluate [tex]\( f(5) \)[/tex] by substituting [tex]\( x = 5 \)[/tex] into the polynomial:
[tex]\[ f(5) = 4(5)^4 - 23(5)^3 + 6(5)^2 + 43(5) + 10. \][/tex]
We calculate each term separately:
1. [tex]\( 4(5)^4 = 4 \times 625 = 2500 \)[/tex]
2. [tex]\( 23(5)^3 = 23 \times 125 = 2875 \)[/tex]
3. [tex]\( 6(5)^2 = 6 \times 25 = 150 \)[/tex]
4. [tex]\( 43(5) = 215 \)[/tex]
5. The constant term is [tex]\( 10 \)[/tex].
Now, let's substitute these back into the expression for [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = 2500 - 2875 + 150 + 215 + 10. \][/tex]
Now, perform the arithmetic:
[tex]\[ f(5) = 2500 - 2875 + 150 + 215 + 10 = 0. \][/tex]
Since [tex]\( f(5) = 0 \)[/tex], it confirms that [tex]\( x - 5 \)[/tex] is indeed a factor of the polynomial [tex]\( f(x) \)[/tex].
In this case, we need to check if [tex]\( f(5) = 0 \)[/tex].
Let's evaluate [tex]\( f(5) \)[/tex] by substituting [tex]\( x = 5 \)[/tex] into the polynomial:
[tex]\[ f(5) = 4(5)^4 - 23(5)^3 + 6(5)^2 + 43(5) + 10. \][/tex]
We calculate each term separately:
1. [tex]\( 4(5)^4 = 4 \times 625 = 2500 \)[/tex]
2. [tex]\( 23(5)^3 = 23 \times 125 = 2875 \)[/tex]
3. [tex]\( 6(5)^2 = 6 \times 25 = 150 \)[/tex]
4. [tex]\( 43(5) = 215 \)[/tex]
5. The constant term is [tex]\( 10 \)[/tex].
Now, let's substitute these back into the expression for [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = 2500 - 2875 + 150 + 215 + 10. \][/tex]
Now, perform the arithmetic:
[tex]\[ f(5) = 2500 - 2875 + 150 + 215 + 10 = 0. \][/tex]
Since [tex]\( f(5) = 0 \)[/tex], it confirms that [tex]\( x - 5 \)[/tex] is indeed a factor of the polynomial [tex]\( f(x) \)[/tex].
