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Answer :
To determine which algebraic expression is a polynomial with a degree of 4, we should first understand what a polynomial is. A polynomial is an expression made up of variables (like x) and coefficients, involving only operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
The degree of a polynomial is the highest power of the variable in the expression. Let's analyze each option step by step to find out which one has a degree of 4:
1. Expression: [tex]\(5x^4 + \sqrt{4}\)[/tex]
- The highest power of [tex]\(x\)[/tex] is [tex]\(4\)[/tex] from the term [tex]\(5x^4\)[/tex].
- The square root [tex]\( \sqrt{4} \)[/tex] is actually a constant ([tex]\(2\)[/tex]) and does not affect the degree.
- Therefore, this expression is a polynomial of degree 4.
2. Expression: [tex]\(x^5 - 6x^4 + 14x^3 + x^2\)[/tex]
- The terms have variable powers of [tex]\(5, 4, 3, \)[/tex] and [tex]\(2\)[/tex].
- The highest power is [tex]\(5\)[/tex] from the term [tex]\(x^5\)[/tex].
- Thus, this expression is a polynomial of degree 5.
3. Expression: [tex]\(9x^4 - x\)[/tex]
- The terms have variable powers of [tex]\(4\)[/tex] and [tex]\(1\)[/tex].
- The highest power is [tex]\(4\)[/tex] from the term [tex]\(9x^4\)[/tex].
- So, this expression is a polynomial of degree 4.
4. Expression: [tex]\(2x^4 - 6x^4 + \frac{14}{x}\)[/tex]
- The terms [tex]\(2x^4\)[/tex] and [tex]\(-6x^4\)[/tex] simplify to [tex]\(0\)[/tex] when combined, so they contribute nothing to the degree.
- The term [tex]\(\frac{14}{x}\)[/tex] is the same as [tex]\(14x^{-1}\)[/tex], which makes this not a polynomial due to the negative exponent.
- Therefore, this expression is not a polynomial.
After evaluating these expressions, the algebraic expressions that are polynomials with a degree of 4 are:
- [tex]\(5x^4 + \sqrt{4}\)[/tex]
- [tex]\(9x^4 - x\)[/tex]
These expressions are both valid polynomials of degree 4, considering the definition and the highest term's power in each expression.
The degree of a polynomial is the highest power of the variable in the expression. Let's analyze each option step by step to find out which one has a degree of 4:
1. Expression: [tex]\(5x^4 + \sqrt{4}\)[/tex]
- The highest power of [tex]\(x\)[/tex] is [tex]\(4\)[/tex] from the term [tex]\(5x^4\)[/tex].
- The square root [tex]\( \sqrt{4} \)[/tex] is actually a constant ([tex]\(2\)[/tex]) and does not affect the degree.
- Therefore, this expression is a polynomial of degree 4.
2. Expression: [tex]\(x^5 - 6x^4 + 14x^3 + x^2\)[/tex]
- The terms have variable powers of [tex]\(5, 4, 3, \)[/tex] and [tex]\(2\)[/tex].
- The highest power is [tex]\(5\)[/tex] from the term [tex]\(x^5\)[/tex].
- Thus, this expression is a polynomial of degree 5.
3. Expression: [tex]\(9x^4 - x\)[/tex]
- The terms have variable powers of [tex]\(4\)[/tex] and [tex]\(1\)[/tex].
- The highest power is [tex]\(4\)[/tex] from the term [tex]\(9x^4\)[/tex].
- So, this expression is a polynomial of degree 4.
4. Expression: [tex]\(2x^4 - 6x^4 + \frac{14}{x}\)[/tex]
- The terms [tex]\(2x^4\)[/tex] and [tex]\(-6x^4\)[/tex] simplify to [tex]\(0\)[/tex] when combined, so they contribute nothing to the degree.
- The term [tex]\(\frac{14}{x}\)[/tex] is the same as [tex]\(14x^{-1}\)[/tex], which makes this not a polynomial due to the negative exponent.
- Therefore, this expression is not a polynomial.
After evaluating these expressions, the algebraic expressions that are polynomials with a degree of 4 are:
- [tex]\(5x^4 + \sqrt{4}\)[/tex]
- [tex]\(9x^4 - x\)[/tex]
These expressions are both valid polynomials of degree 4, considering the definition and the highest term's power in each expression.
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