Q. ( -frac{p}{24} x^4 + frac{f}{6} x^3 + a x + b ).

Expression

\(-\frac{p}{24}x^4+\frac{f}{6}x^3+ax+b\)

This is a polynomial expression in the variable \(x\). The letters \(p\), \(f\), \(a\), and \(b\) are constants, so they act like fixed numbers.

Step 1: Identify each term

The expression has four separate terms.

The first term is \(-\frac{p}{24}x^4\).

The second term is \(\frac{f}{6}x^3\).

The third term is \(ax\).

The fourth term is \(b\).

Each term has a coefficient. Every term except \(b\) also has a power of \(x\).

Step 2: Find the degree of the polynomial

The degree of a polynomial is the highest power of \(x\) in the expression.

The powers of \(x\) in this expression are \(x^4\), \(x^3\), \(x^1\), and \(x^0\).

The highest power is \(4\), so the expression is a fourth-degree polynomial.

\(\text{Degree}=4\)

Step 3: Identify the leading term

The leading term is the term with the highest power of \(x\).

The highest power is \(x^4\), so the leading term is \(-\frac{p}{24}x^4\).

The leading coefficient is the constant part multiplied by \(x^4\).

So, the leading coefficient is \(-\frac{p}{24}\).

Step 4: Identify the constant term

The constant term is the term without \(x\).

Here, the constant term is \(b\).

Step 5: Check whether the expression can be simplified

To simplify a polynomial, combine like terms. Like terms must have the same power of \(x\).

This expression has terms with different powers of \(x\): \(x^4\), \(x^3\), \(x^1\), and \(x^0\).

No two terms have the same power of \(x\), so there are no like terms to combine.

Therefore, the expression is already simplified.

Final Answer

The expression is already in simplified polynomial form. It is a fourth-degree polynomial in \(x\). Its leading term is \(-\frac{p}{24}x^4\), its leading coefficient is \(-\frac{p}{24}\), and its constant term is \(b\).

\(-\frac{p}{24}x^4+\frac{f}{6}x^3+ax+b\)