Let `P(x) =ax^(7) + bx^(3) +cx-5`, where a,b,c are constants. Given `P(-7) =7`, find the value of P(7).

Answer

To solve the problem, we start with the polynomial given:

\[ P(x) = ax^7 + bx^3 + cx - 5 \]

We know that:

\[ P(-7) = 7 \]

### Step 1: Substitute -7 into P(x)

We will substitute \( x = -7 \) into the polynomial:

\[ P(-7) = a(-7)^7 + b(-7)^3 + c(-7) - 5 \]

### Step 2: Calculate the powers of -7

Calculating the powers:

\[ (-7)^7 = -7^7 \quad \text{(since odd power of negative number is negative)} \]
\[ (-7)^3 = -7^3 \quad \text{(same reasoning)} \]
\[ (-7) = -7 \]

So we can rewrite \( P(-7) \):

\[ P(-7) = -a(7^7) - b(7^3) - 7c - 5 \]

### Step 3: Set the equation equal to 7

From the problem, we know:

\[ -a(7^7) - b(7^3) - 7c - 5 = 7 \]

### Step 4: Rearrange the equation

Rearranging gives us:

\[ -a(7^7) - b(7^3) - 7c = 7 + 5 \]
\[ -a(7^7) - b(7^3) - 7c = 12 \]

### Step 5: Find P(7)

Now we need to find \( P(7) \):

\[ P(7) = a(7^7) + b(7^3) + c(7) - 5 \]

### Step 6: Relate P(7) to P(-7)

Notice that:

\[ P(7) = -P(-7) + 12 - 5 \]

Since we have \( P(-7) = 7 \):

\[ P(7) = -7 + 12 - 5 \]

### Step 7: Simplify

Now we simplify:

\[ P(7) = -7 + 12 - 5 = 0 \]

Thus, the value of \( P(7) \) is:

\[ \boxed{0} \]