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Sagot :
To determine the value of [tex]\( m \)[/tex] in the given expression [tex]\( 4 x^5 y^m z \)[/tex], we need to understand the concept of the degree of a polynomial.
The degree of a polynomial is the sum of the exponents of the variables in each term, and for a single term polynomial, it is simply the sum of the exponents of the variables in that term.
Let's analyze the term [tex]\( 4 x^5 y^m z \)[/tex]:
- The exponent of [tex]\( x \)[/tex] is [tex]\( 5 \)[/tex].
- The exponent of [tex]\( y \)[/tex] is [tex]\( m \)[/tex].
- The exponent of [tex]\( z \)[/tex] is [tex]\( 1 \)[/tex].
The degree of the expression is given to be [tex]\( 10 \)[/tex]. Hence, we can write:
[tex]\[ \text{Degree of the expression} = \text{Exponent of } x + \text{Exponent of } y + \text{Exponent of } z \][/tex]
[tex]\[ 10 = 5 + m + 1 \][/tex]
Now, solve for [tex]\( m \)[/tex]:
[tex]\[ 10 = 5 + m + 1 \][/tex]
[tex]\[ 10 = 6 + m \][/tex]
[tex]\[ m = 10 - 6 \][/tex]
[tex]\[ m = 4 \][/tex]
Hence, the value of [tex]\( m \)[/tex] is [tex]\( 4 \)[/tex].
The degree of a polynomial is the sum of the exponents of the variables in each term, and for a single term polynomial, it is simply the sum of the exponents of the variables in that term.
Let's analyze the term [tex]\( 4 x^5 y^m z \)[/tex]:
- The exponent of [tex]\( x \)[/tex] is [tex]\( 5 \)[/tex].
- The exponent of [tex]\( y \)[/tex] is [tex]\( m \)[/tex].
- The exponent of [tex]\( z \)[/tex] is [tex]\( 1 \)[/tex].
The degree of the expression is given to be [tex]\( 10 \)[/tex]. Hence, we can write:
[tex]\[ \text{Degree of the expression} = \text{Exponent of } x + \text{Exponent of } y + \text{Exponent of } z \][/tex]
[tex]\[ 10 = 5 + m + 1 \][/tex]
Now, solve for [tex]\( m \)[/tex]:
[tex]\[ 10 = 5 + m + 1 \][/tex]
[tex]\[ 10 = 6 + m \][/tex]
[tex]\[ m = 10 - 6 \][/tex]
[tex]\[ m = 4 \][/tex]
Hence, the value of [tex]\( m \)[/tex] is [tex]\( 4 \)[/tex].
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