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