If (x 2- 1) is a factor of ax 4+ bx 3+ cx 2+ dx + e, then which one of the following is correct?
a + c + e = b + d
Understanding the Problem: Polynomial Factors and Coefficients
The question asks for the relationship between the coefficients (\(a, b, c, d, e\)) of a polynomial \(P(x) = ax^4 + bx^3 + cx^2 + dx + e\) when we are given that \( (x^2 - 1) \) is a factor of this polynomial.
When a polynomial \(P(x)\) has a factor \( (x^2 - 1) \), it means that \( P(x) \) can be divided by \( (x^2 - 1) \) with a remainder of zero. This property is closely related to the roots of the factor.
Applying the Factor Theorem
The Factor Theorem states that if \( (x - k) \) is a factor of a polynomial \( P(x) \), then \( P(k) = 0 \). Similarly, if \( P(k) = 0 \), then \( (x - k) \) is a factor of \( P(x) \).
In this case, the factor is \( (x^2 - 1) \). We can factorize \( (x^2 - 1) \) as \( (x - 1)(x + 1) \). This means that if \( (x^2 - 1) \) is a factor of \( P(x) \), then both \( (x - 1) \) and \( (x + 1) \) must be factors of \( P(x) \).
According to the Factor Theorem:
- If \( (x - 1) \) is a factor, then \( P(1) \) must be equal to 0.
- If \( (x + 1) \) is a factor, then \( P(-1) \) must be equal to 0.
Calculating P(1) and P(-1)
Let's substitute \( x = 1 \) into the polynomial \( P(x) = ax^4 + bx^3 + cx^2 + dx + e \):
Since \( (x - 1) \) is a factor, \( P(1) = 0 \). So, we get our first equation:
Now, let's substitute \( x = -1 \) into the polynomial \( P(x) = ax^4 + bx^3 + cx^2 + dx + e \):
Since \( (x + 1) \) is a factor, \( P(-1) = 0 \). So, we get our second equation:
Finding the Relationship between Coefficients
We now have two equations based on the condition that \( (x^2 - 1) \) is a factor:
- \( a + b + c + d + e = 0 \)
- \( a - b + c - d + e = 0 \)
We can manipulate these equations to find relationships between the coefficients. Let's add Equation 1 and Equation 2:
Combining like terms:
Dividing by 2:
Now, let's subtract Equation 2 from Equation 1:
Combining like terms:
Dividing by 2:
So, for \( (x^2 - 1) \) to be a factor of the polynomial, it is necessary and sufficient that both \( a + c + e = 0 \) and \( b + d = 0 \).
Checking the Options
We need to see which of the given options is a correct consequence of the conditions \( a + c + e = 0 \) and \( b + d = 0 \).
The conditions are:
- \( a + c + e = 0 \)
- \( b + d = 0 \)
Let's look at Option 4: \( a + c + e = b + d \).
If \( a + c + e = 0 \) and \( b + d = 0 \), then substituting these values into the equation from Option 4 gives:
This statement is true. This means that the relationship \( a + c + e = b + d \) is always true when \( (x^2 - 1) \) is a factor of the polynomial.
Let's briefly look at the other options:
- Option 1: \( a + b + c = d + e \). This can be rewritten as \( a + c - e = d - b \). Using our conditions \( a+c = -e \) and \( d = -b \), this becomes \( -e - e = -b - b \), which simplifies to \( -2e = -2b \) or \( e = b \). This is not always true; \( e \) is not necessarily equal to \( b \).
- Option 2: \( a + b + e = c + d \). This can be rewritten as \( a + e - c = d - b \). Using our conditions \( a+e = -c \) and \( d = -b \), this becomes \( -c - c = -b - b \), which simplifies to \( -2c = -2b \) or \( c = b \). This is not always true; \( c \) is not necessarily equal to \( b \).
- Option 3: \( b + c + d = a + e \). This can be rewritten as \( (b + d) + c = a + e \). Using our conditions \( b+d = 0 \) and \( a+e = -c \), this becomes \( 0 + c = -c \), which simplifies to \( c = -c \) or \( 2c = 0 \), meaning \( c=0 \). This is not always true; \( c \) is not necessarily equal to 0.
Only Option 4, \( a + c + e = b + d \), is a relationship that must hold true when \( (x^2 - 1) \) is a factor of the polynomial \( ax^4 + bx^3 + cx^2 + dx + e \).
| Condition from Factor Theorem | Resulting Relationship |
|---|---|
| \( P(1) = 0 \) | \( a + b + c + d + e = 0 \) |
| \( P(-1) = 0 \) | \( a - b + c - d + e = 0 \) |
| Combination of Equations | Derived Condition |
|---|---|
| (Eq 1) + (Eq 2) | \( a + c + e = 0 \) |
| (Eq 1) - (Eq 2) | \( b + d = 0 \) |
Since \( a + c + e = 0 \) and \( b + d = 0 \), it directly follows that \( a + c + e = b + d \) because both sides of the equation are equal to zero.
Conclusion
Based on the Factor Theorem and the properties of polynomial division, if \( (x^2 - 1) \) is a factor of the given polynomial, the coefficients must satisfy the condition \( a + c + e = b + d \).
Revision Table: Polynomial Factors
| Concept | Description | Relevance Here |
|---|---|---|
| Factor Theorem | \( (x - k) \) is a factor of \( P(x) \) if and only if \( P(k) = 0 \). | Used to determine the conditions \( P(1)=0 \) and \( P(-1)=0 \). |
| Roots of \( x^2 - 1 \) | The roots are \( x=1 \) and \( x=-1 \). | These are the values of \( x \) for which \( P(x) \) must be zero if \( (x^2-1) \) is a factor. |
| Polynomial \( P(x) \) | \( ax^4 + bx^3 + cx^2 + dx + e \) | The polynomial whose coefficients' relationship is being investigated. |
Additional Information: General Case
For a general polynomial \( P(x) \) and a factor \( (x^2 - k^2) \), the roots are \( x=k \) and \( x=-k \). Thus, \( P(k)=0 \) and \( P(-k)=0 \) must hold.
For \( P(x) = ax^4 + bx^3 + cx^2 + dx + e \):
\( P(k) = ak^4 + bk^3 + ck^2 + dk + e = 0 \)
\( P(-k) = a(-k)^4 + b(-k)^3 + c(-k)^2 + d(-k) + e = ak^4 - bk^3 + ck^2 - dk + e = 0 \)
Adding these two equations gives:
\( 2ak^4 + 2ck^2 + 2e = 0 \implies ak^4 + ck^2 + e = 0 \)
Subtracting the second from the first gives:
\( 2bk^3 + 2dk = 0 \implies bk^3 + dk = 0 \implies k(bk^2 + d) = 0 \)
In our specific problem, \( k=1 \). This leads to \( a(1)^4 + c(1)^2 + e = 0 \implies a+c+e=0 \) and \( 1(b(1)^2 + d) = 0 \implies b+d=0 \), which matches our derived conditions.
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