If (x + 2) is a common factor of x 2+ ax + b and x 2+ bx + a, then the ratio of a ∶ b is equal to
1
Finding the Ratio of Coefficients Using the Factor Theorem
The problem states that the expression \((x + 2)\) is a common factor of two quadratic polynomials: \(x^2 + ax + b\) and \(x^2 + bx + a\). We are asked to find the ratio of the coefficients \(a\) and \(b\), represented as \(a \ratio b\).
According to the Factor Theorem, if \((x - c)\) is a factor of a polynomial \(P(x)\), then \(P(c) = 0\). In this case, the common factor is \((x + 2)\), which can be written as \((x - (-2))\). Therefore, if \((x + 2)\) is a factor of a polynomial, substituting \(x = -2\) into the polynomial must result in zero.
Applying the Factor Theorem to the First Polynomial
Let the first polynomial be \(P(x) = x^2 + ax + b\). Since \((x + 2)\) is a factor, we have:
\[P(-2) = (-2)^2 + a(-2) + b = 0\] \[4 - 2a + b = 0\]Rearranging this equation, we get our first relationship between \(a\) and \(b\):
\[2a - b = 4 \quad \cdots (1)\]Applying the Factor Theorem to the Second Polynomial
Let the second polynomial be \(Q(x) = x^2 + bx + a\). Since \((x + 2)\) is also a factor of this polynomial, we have:
\[Q(-2) = (-2)^2 + b(-2) + a = 0\] \[4 - 2b + a = 0\]Rearranging this equation, we get our second relationship between \(a\) and \(b\):
\[a - 2b = -4 \quad \cdots (2)\]Solving the System of Linear Equations
Now we have a system of two linear equations with two variables, \(a\) and \(b\):
- \(2a - b = 4\)
- \(a - 2b = -4\)
We can solve this system using methods like substitution or elimination. Let's use the elimination method. Multiply equation (2) by 2:
\[2(a - 2b) = 2(-4)\] \[2a - 4b = -8 \quad \cdots (3)\]Now, subtract equation (3) from equation (1):
\[(2a - b) - (2a - 4b) = 4 - (-8)\] \[2a - b - 2a + 4b = 4 + 8\] \[3b = 12\]Solving for \(b\):
\[b = \frac{12}{3}\] \[b = 4\]Now substitute the value of \(b\) back into either equation (1) or (2) to find \(a\). Using equation (1):
\[2a - (4) = 4\] \[2a - 4 = 4\] \[2a = 4 + 4\] \[2a = 8\]Solving for \(a\):
\[a = \frac{8}{2}\] \[a = 4\]Determining the Ratio a : b
We found that \(a = 4\) and \(b = 4\). The ratio \(a \ratio b\) is therefore:
\[a \ratio b = 4 \ratio 4\]This ratio can be simplified by dividing both sides by the greatest common divisor, which is 4:
\[a \ratio b = \frac{4}{4} \ratio \frac{4}{4}\] \[a \ratio b = 1 \ratio 1\]Conclusion
If \((x + 2)\) is a common factor of \(x^2 + ax + b\) and \(x^2 + bx + a\), then the ratio of \(a\) to \(b\) is \(1 \ratio 1\). This means \(a\) is equal to \(b\).
| Polynomial | Condition from Factor Theorem | Equation Derived |
|---|---|---|
| \(x^2 + ax + b\) | Substitute \(x=-2\) | \(4 - 2a + b = 0 \implies 2a - b = 4\) |
| \(x^2 + bx + a\) | Substitute \(x=-2\) | \(4 - 2b + a = 0 \implies a - 2b = -4\) |
Revision Table: Key Concepts Reviewed
| Concept | Description | Application in Problem |
|---|---|---|
| Factor Theorem | If \((x - c)\) is a factor of \(P(x)\), then \(P(c) = 0\). | Used to set \(x = -2\) in both polynomials. |
| Root of a Polynomial | A value \(c\) such that \(P(c) = 0\). If \((x-c)\) is a factor, then \(c\) is a root. | \(x = -2\) is a common root for both polynomials. |
| Simultaneous Equations | A set of equations with multiple variables, solved together to find values that satisfy all equations. | The two equations derived from the Factor Theorem were solved to find \(a\) and \(b\). |
| Ratio | A comparison of two quantities. Represented as \(a \ratio b\) or \(\frac{a}{b}\). | Calculated the ratio of the determined values of \(a\) and \(b\). |
Additional Information: Understanding Polynomial Factors
A factor of a polynomial is an expression that divides the polynomial exactly, leaving no remainder. The Factor Theorem provides a direct link between the roots of a polynomial and its linear factors. If you know a root \(c\), you know \((x-c)\) is a factor, and vice versa.
In this problem, since \((x+2)\) is a common factor, it means \(x=-2\) is a common root of both quadratic equations \(x^2 + ax + b = 0\) and \(x^2 + bx + a = 0\). This property is what allows us to substitute \(x=-2\) and form equations to solve for the unknown coefficients \(a\) and \(b\). Common factors play an important role in simplifying polynomial expressions and finding their roots.
Understanding the relationship between factors, roots, and coefficients is fundamental in algebra. Problems like this reinforce the application of key theorems such as the Factor Theorem and techniques for solving systems of equations.
Similar Questions
The cost of diamond varies directly as the square of its weight. A diamond broke into four pieces with their weights in the ratio of 1 ∶ 2 ∶ 3 ∶ 4. If the loss in total value of the diamond was Rs. 70000, what was the price of the original diamond?
25 kg of alloy X is mixed with 125 kg of alloy Y. If the amount of lead and tin in the alloy X is in the ratio 1 ∶ 2 and the amount of lead and tin in the alloy Y is in the ratio 2 ∶ 3, then what is the ratio of lead and tin in the mixture?
If \(\frac{{4a\; + \;7b}}{{4c\; +\; 7d}} = \frac{{4a\; - \;7b}}{{4c\; -\; 7d}}\) , then which one of the following is correct?
If a: b = c : d = 1 : 6, then what is the value of \(\frac{{{a^2}\; + \;{c^2}}}{{{b^2}\; + \;{d^2}}}?\)
Important Questions from Compound Ratios
- What is the compound ratio of $45 : 75$, $3 : 5$, $51 : 68$ and $256 : 81$?
If A : B = 2 : 5, B : C = 4 : 3 and C : D = 2 : 1, then what is value of A : C : D?
The following ratios are given:
\(a : b = 2 : 3\)
\(b : c = 4 : 5\)
\(c : d = 3 : 2\)
\(d : e = 5 : 4\)
What is the value of the ratio a : e?
Rs. 2420 were divided among A, B, C so that A : B = 5 : 4 and B : C = 9 : 10 then C gets:
The sum of three numbers is 280. If the ratio between the first and second numbers is 2 : 3 and the ratio between second and third numbers is 4 : 5, find the second number.
