Question

Find a polynomial \(f\) that satisfies the following properties. (Hint: Determine the degree of \(f ;\) then substitute a polynomial of that degree and solve for its coefficients.) $$(f(x))^{2}=9 x^{2}-12 x+4$$

Short answer

Answer: The possible polynomial functions f(x) are either \(f(x) = 3x - 2\) or \(f(x) = -3x + 2\).

Step-by-step solution

Determine the degree of f(x)

We can determine the degree of f(x) by looking at the given equation \((f(x))^2 = 9x^2 - 12x + 4\). Since the highest power in the equation is \(x^2\), the degree of \((f(x))^2\) is 2. Since the degree of a polynomial squared is double the degree of the polynomial itself, we can conclude that the degree of f(x) is 1.

Write the general form of f(x)

Since f(x) is a polynomial function of degree 1, we can write its general form as: $$f(x) = ax + b$$ where a and b are coefficients to be determined.

Substitute the general form into the equation and expand

Replace f(x) with the general form of f(x) in the given equation and expand: $$(ax + b)^2 = 9x^2 - 12x + 4$$

Simplify and equate coefficients

We can simplify the equation and then equate the coefficients on both sides: $$a^2x^2 + 2abx + b^2 = 9x^2 - 12x + 4$$ By equating the coefficients of the same powers of x, we have: $$a^2 = 9$$ $$2ab = -12$$ $$b^2 = 4$$

Solve for coefficients a and b

Now, solve the three equations for a and b: From \(a^2 = 9\), we get \(a = \pm 3\). From \(b^2 = 4\), we get \(b = \pm 2\). Next, we have to solve the equation \(2ab = -12\). There are four cases to consider to find possible values for a and b. Case 1: a = 3 and b = 2 $$2(3)(2) = 12 \neq -12$$, so it's not a valid solution. Case 2: a = 3 and b = -2 $$2(3)(-2) = -12$$, so a = 3 and b = -2 is a valid solution. Case 3: a = -3 and b = 2 $$2(-3)(2) = -12$$, so a = -3 and b = 2 is a valid solution. Case 4: a = -3 and b = -2 $$2(-3)(-2) = 12 \neq -12$$, so it's not a valid solution.

Write the final polynomial f(x)

We have two valid solutions for the coefficients a and b: 1. a = 3, b = -2, so \(f(x) = 3x - 2\) 2. a = -3, b = 2, so \(f(x) = -3x + 2\) Therefore, the polynomial f(x) can be either \(f(x) = 3x - 2\) or \(f(x) = -3x + 2\).

Key concepts

Degree of a Polynomial

The degree of a polynomial is a key concept in understanding how these mathematical expressions behave. Simply put, the degree is the highest power of the variable present in the polynomial. For instance, in the polynomial \(x^2 + 3x + 5\), the degree is 2 because the highest power of \(x\) is 2.
In the exercise given, we started with the equation \((f(x))^2 = 9x^2 - 12x + 4\). Here, \(x^2\) is the term with the highest power, making the degree of \((f(x))^2\) 2. Since this polynomial is squared, the degree of \(f(x)\) must be half of this, resulting in a degree of 1 for \(f(x)\).
This understanding helps us know that \(f(x)\) is a linear polynomial, meaning it has one variable with no higher powers, and we can express it as \(f(x) = ax + b\). Identifying the degree is crucial because it sets the limits for the form of \(f(x)\).

Coefficients of Polynomials

Coefficients are the numerical factors in front of the variables in a polynomial and play a vital role in shaping the polynomial's graph and other properties. In our case with \(f(x) = ax + b\), \(a\) and \(b\) are coefficients.
To find these coefficients, we substitute the polynomial into the given equation and expand: \((ax + b)^2 = a^2x^2 + 2abx + b^2\). This step is necessary to equate it with \(9x^2 - 12x + 4\) and find the matching coefficients. By doing this comparison:

• \(a^2\) should match the coefficient of \(x^2\), which is 9.
• \(2ab\) should match the coefficient of \(x\), which is -12.
• \(b^2\) should match the constant term, which is 4.

With these equations, we determine \(a\) as \(\pm 3\) and \(b\) as \(\pm 2\). Re-evaluating these values through the equation \(2ab = -12\) helps us find the correct combinations of \(a\) and \(b\) that work. Coefficients like \(a\) and \(b\) are essential for both forming the polynomial and understanding its graph's features like direction and steepness.

Solving Equations

Solving polynomial equations involves finding the values of variables that satisfy the equation. The exercise involves manipulating equations to find the correct coefficients to form the polynomial \(f(x)\).
Once we identified our coefficients' relationships, through equations like \(a^2 = 9\), \(b^2 = 4\), and \(2ab = -12\), we have four potential cases for \(a\) and \(b\). Our goal is to find which combinations are valid:

• \(a = 3\) and \(b = 2\) results in \(2(3)(2) = 12\), which does not satisfy \(2ab = -12\).
• \(a = 3\) and \(b = -2\) results in \(2(3)(-2) = -12\), which is valid.
• \(a = -3\) and \(b = 2\) results in \(2(-3)(2) = -12\), also valid.
• \(a = -3\) and \(b = -2\) results in \(2(-3)(-2) = 12\), not valid.

Thus, the valid solutions are \(f(x) = 3x - 2\) and \(f(x) = -3x + 2\). Solving equations like this involves systematically testing potential solutions and ensuring all conditions are met, allowing us to confidently determine the correct polynomial functions that meet the requirements.