Questions: Consider the algebraic expression sqrt(7) x^19 + 13.7 x^14 + (pi/2) x^7 + 1/8. What is the degree of this polynomial? Identify the constant term. Identify the leading coefficient. Identify the leading term.

Solution Steps

1. Degree of the Polynomial: The degree of a polynomial is the highest power of the variable $x$ in the expression.
2. Constant Term: The constant term is the term in the polynomial that does not contain any variables.
3. Leading Coefficient: The leading coefficient is the coefficient of the term with the highest power of $x$.
4. Leading Term: The leading term is the term with the highest power of $x$.

The degree of the polynomial $\sqrt{7} x^{19} + 13.7 x^{14} + \frac{\pi}{2} x^{7} + \frac{1}{8}$ is determined by the highest power of $x$. In this case, the highest power is $19$. Therefore, the degree is: $$\text{Degree} = 19$$

The constant term in the polynomial is the term that does not contain the variable $x$. From the expression, the constant term is: $$\text{Constant Term} = \frac{1}{8} = 0.125$$

The leading coefficient is the coefficient of the term with the highest power of $x$. For the term $\sqrt{7} x^{19}$, the leading coefficient is: $$\text{Leading Coefficient} = \sqrt{7} \approx 2.6458$$

The leading term is the term with the highest power of $x$, which is: $$\text{Leading Term} = \sqrt{7} x^{19}$$

Final Answer