To determine if [tex]\( x = 1 \)[/tex] is a root of the polynomial [tex]\( 3x^3 - 4x^2 + 3x - 7 \)[/tex], we need to verify if substituting [tex]\( x = 1 \)[/tex] into the polynomial yields a result of zero.
1. First, consider the polynomial:
[tex]\[ P(x) = 3x^3 - 4x^2 + 3x - 7 \][/tex]
2. Substitute [tex]\( x = 1 \)[/tex] into the polynomial:
[tex]\[ P(1) = 3(1)^3 - 4(1)^2 + 3(1) - 7 \][/tex]
3. Simplify each term individually:
[tex]\[ 3(1)^3 = 3 \][/tex]
[tex]\[ -4(1)^2 = -4 \][/tex]
[tex]\[ +3(1) = 3 \][/tex]
[tex]\[ -7 = -7 \][/tex]
4. Combine these results:
[tex]\[ P(1) = 3 - 4 + 3 - 7 \][/tex]
5. Simplify the expression step by step:
[tex]\[ 3 - 4 = -1 \][/tex]
[tex]\[ -1 + 3 = 2 \][/tex]
[tex]\[ 2 - 7 = -5 \][/tex]
6. Since the result is [tex]\( P(1) = -5 \)[/tex] and not zero, [tex]\( x = 1 \)[/tex] is not a root of the polynomial.
However, considering the given result, the proper interpretation might be to examine the result itself rather than proving it directly. Thus, it has been demonstrated numerically that substituting [tex]\( x = 1 \)[/tex] into [tex]\( 3x^3 - 4x^2 + 3x - 7 \)[/tex] results in [tex]\( -5 \)[/tex], confirming that [tex]\( x = 1 \)[/tex] does not satisfy [tex]\( P(x) = 0 \)[/tex].
Therefore, [tex]\( x = 1 \)[/tex] is not a root of the polynomial [tex]\( 3x^3 - 4x^2 + 3x - 7 \)[/tex].
