What is the degree and leading coefficient of (2x−3)7. Find the coefficient of x4 (Hint: use Pascal’s triangle).

To solve this problem, we need to expand \((2x - 3)^7\) using the binomial theorem and then extract the relevant information for the degree, leading coefficient, and the coefficient of \(x^4\).

### Step 1: Degree and Leading Coefficient
The degree of a polynomial is the highest power of the variable \(x\) in the expansion. The expression \((2x - 3)^7\) is a binomial raised to the 7th power. The highest power of \(x\) will be \(x^7\), so the degree of the polynomial is 7.

The leading term comes from the first term of the binomial expansion where the entire power is on \(2x\). This term is:
\[
\binom{7}{0} (2x)^7 (-3)^0 = 2^7 x^7 = 128x^7
\]
So, the leading coefficient is **128**.

### Step 2: Coefficient of \(x^4\)
To find the coefficient of \(x^4\), we need to apply the binomial expansion:
\[
(2x - 3)^7 = \sum_{k=0}^{7} \binom{7}{k} (2x)^{7-k} (-3)^k
\]
We are looking for the term where \(x^4\) appears. This happens when \((2x)^{4}\), which corresponds to the term where \(k = 3\), because:
\[
(2x)^{7-k} = (2x)^4 \quad \text{when} \quad k = 3
\]
Now, compute the term:
\[
\binom{7}{3} (2x)^4 (-3)^3
\]
First, find \(\binom{7}{3}\):
\[
\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]
Now calculate the other parts:
\[
(2x)^4 = 16x^4 \quad \text{and} \quad (-3)^3 = -27
\]
So, the term is:
\[
35 \times 16x^4 \times (-27) = 35 \times 16 \times (-27)x^4 = -15120x^4
\]
Therefore, the coefficient of \(x^4\) is **-15120**.

### Final Answer:
- The degree of the polynomial is **7**.
- The leading coefficient is **128**.
- The coefficient of \(x^4\) is **-15120**.

Would you like more details on this solution, or do you have any specific questions?

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Here are 5 related questions:
1. How do you use Pascal's triangle to expand a binomial expression?
2. What is the general form of the binomial theorem?
3. How can you find the coefficient of any term in a binomial expansion?
4. How do the signs alternate in binomial expansions involving subtraction?
5. What are the applications of the binomial theorem in calculus?

**Tip:** When expanding binomials, using Pascal's triangle can simplify finding the binomial coefficients without calculations.

What is the degree and leading coefficient of (2x−3)^7. Find the coefficient of x^4 (Hint: use Pascal’s triangle).

Discover how to find the degree, leading coefficient, and coefficient of x^4 in the expansion of (2x−3)^7 using the binomial theorem and Pascal’s triangle. This problem involves polynomial expansion and algebraic manipulation, suitable for Grades 10-12.

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