Consider the following statements in respect of the polynomial a(b - c) (x - b) (x - c) + b(c - a) (x - c) (x - a) + c(a - b) (x - a) (x - b): 1. The coefficient of x2 is 0. 2. The coefficient of x is (a - b) (b - c) (c - a). Which of the statements given above is/are correct ?
1 only
Analyzing Polynomial Coefficients
We are given a polynomial and asked to determine the correctness of two statements regarding its coefficients: the coefficient of \(x^2\) and the coefficient of \(x\). Let's expand the given polynomial term by term to find these coefficients.
The polynomial is: \(a(b - c) (x - b) (x - c) + b(c - a) (x - c) (x - a) + c(a - b) (x - a) (x - b)\)
Let's expand each term:
First term: \(a(b - c) (x - b) (x - c)\)
First, expand \((x - b) (x - c) = x^2 - cx - bx + bc = x^2 - (b+c)x + bc\).
So, the first term is \(a(b - c) (x^2 - (b+c)x + bc) = a(b - c)x^2 - a(b - c)(b+c)x + a(b - c)bc\).
Second term: \(b(c - a) (x - c) (x - a)\)
Expand \((x - c) (x - a) = x^2 - ax - cx + ca = x^2 - (c+a)x + ca\).
So, the second term is \(b(c - a) (x^2 - (c+a)x + ca) = b(c - a)x^2 - b(c - a)(c+a)x + b(c - a)ca\).
Third term: \(c(a - b) (x - a) (x - b)\)
Expand \((x - a) (x - b) = x^2 - bx - ax + ab = x^2 - (a+b)x + ab\).
So, the third term is \(c(a - b) (x^2 - (a+b)x + ab) = c(a - b)x^2 - c(a - b)(a+b)x + c(a - b)ab\).
Calculating the Coefficient of \(x^2\)
The coefficient of \(x^2\) in the entire polynomial is the sum of the coefficients of \(x^2\) from each expanded term:
Coefficient of \(x^2 = a(b - c) + b(c - a) + c(a - b)\)
Let's expand this expression:
\(ab - ac + bc - ba + ca - cb\)
Group terms with the same variables:
\((ab - ba) + (-ac + ca) + (bc - cb) = 0 + 0 + 0 = 0\).
So, the coefficient of \(x^2\) is 0.
Statement 1: The coefficient of \(x^2\) is 0. This statement is correct.
Calculating the Coefficient of \(x\)
The coefficient of \(x\) in the entire polynomial is the sum of the coefficients of \(x\) from each expanded term:
Coefficient of \(x = -a(b - c)(b + c) - b(c - a)(c + a) - c(a - b)(a + b)\)
Using the difference of squares formula, \((x-y)(x+y) = x^2 - y^2\):
\(-a(b^2 - c^2) - b(c^2 - a^2) - c(a^2 - b^2)\)
Expand this expression:
\(-ab^2 + ac^2 - bc^2 + ba^2 - ca^2 + cb^2\)
Rearrange the terms by \(a^2, b^2, c^2\):
\(a^2(b - c) + b^2(c - a) + c^2(a - b)\)
This is a known cyclic expression. Let's compare it to the expression given in Statement 2: \((a - b)(b - c)(c - a)\).
Let's expand \((a - b)(b - c)(c - a)\):
\((ab - ac - b^2 + bc)(c - a)\)
\(abc - a^2b - ac^2 + a^2c - b^2c + ab^2 + bc^2 - abc\)
Combine like terms:
\(-a^2b + a^2c - ac^2 - b^2c + ab^2 + bc^2\)
Rearrange:
\(a^2(c - b) + b^2(a - c) + c^2(b - a)\)
This is equal to \(-(a^2(b - c) + b^2(c - a) + c^2(a - b))\).
So, the coefficient of \(x\) is \(a^2(b - c) + b^2(c - a) + c^2(a - b)\), which is equal to \(-(a - b)(b - c)(c - a)\).
Statement 2: The coefficient of \(x\) is \((a - b) (b - c) (c - a)\). This statement is incorrect.
Summary of Findings
- Statement 1 about the coefficient of \(x^2\) is correct.
- Statement 2 about the coefficient of \(x\) is incorrect.
Therefore, only Statement 1 is correct.
Revision Table: Polynomial Coefficients
Let's summarize the key calculations for the polynomial coefficients.
| Term | Expanded Form (relevant parts) | Coefficient of \(x^2\) | Coefficient of \(x\) |
|---|---|---|---|
| \(a(b - c) (x - b) (x - c)\) | \(a(b - c)x^2 - a(b - c)(b+c)x + \dots\) | \(a(b - c)\) | \(-a(b^2 - c^2)\) |
| \(b(c - a) (x - c) (x - a)\) | \(b(c - a)x^2 - b(c - a)(c+a)x + \dots\) | \(b(c - a)\) | \(-b(c^2 - a^2)\) |
| \(c(a - b) (x - a) (x - b)\) | \(c(a - b)x^2 - c(a - b)(a+b)x + \dots\) | \(c(a - b)\) | \(-c(a^2 - b^2)\) |
| Total Polynomial | Sum of terms | \(a(b - c) + b(c - a) + c(a - b) = 0\) | \(-(ab^2 - ac^2 + bc^2 - ba^2 + ca^2 - cb^2) = -(a^2(b - c) + b^2(c - a) + c^2(a - b))\) |
The coefficient of \(x^2\) is confirmed to be 0. The coefficient of \(x\) is \(a^2(b - c) + b^2(c - a) + c^2(a - b)\). The statement claims it is \((a - b)(b - c)(c - a)\), which is equal to \(-(a^2(b - c) + b^2(c - a) + c^2(a - b))\). These two expressions have opposite signs, so Statement 2 is incorrect.
Additional Information on Polynomial Expansion
Expanding polynomials is a fundamental skill in algebra. To find coefficients, you can multiply out all terms or use distributive properties carefully. For expressions with multiple factors like \((x-b)(x-c)\), multiplying two binomials first simplifies the process. The general form of a quadratic is \(Ax^2 + Bx + C\), where A is the coefficient of \(x^2\), B is the coefficient of \(x\), and C is the constant term.
In this specific polynomial, a key observation for experienced students might be that the polynomial evaluates to 0 when \(x=a\), \(x=b\), or \(x=c\). For example, if we substitute \(x=a\): The first term becomes \(a(b-c)(a-b)(a-c)\). The second term becomes \(b(c-a)(a-c)(a-a) = 0\). The third term becomes \(c(a-b)(a-a)(a-b) = 0\). So the total polynomial is \(a(b-c)(a-b)(a-c)\) when \(x=a\). This is not necessarily zero unless \(a, b, c\) are such that this evaluates to zero. Let's re-check the substitution approach carefully.
Let \(P(x) = a(b - c) (x - b) (x - c) + b(c - a) (x - c) (x - a) + c(a - b) (x - a) (x - b)\).
If \(x=a\): \(P(a) = a(b - c) (a - b) (a - c) + b(c - a) (a - c) (a - a) + c(a - b) (a - a) (a - b)\) \(P(a) = a(b - c) (a - b) (a - c) + b(c - a) (a - c) (0) + c(a - b) (0) (a - b)\) \(P(a) = a(b - c) (a - b) (a - c)\). This does not necessarily mean the polynomial is zero at \(x=a, b, c\). The structure resembles Lagrange interpolation, but it's not quite the same form.
The direct expansion method used earlier is reliable for finding the exact coefficients.
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