Question

Suppose \(f(x)=2 x^{3}-14 x^{2}+b x-3\) with \(f(2)=0\) and \(g(x)=x^{3}+c x^{2}-8 x+30,\) with the zero \(x=3-i,\) where \(b\) and \(c\) are real numbers. Find \((f \cdot g)(1) .\)

Short answer

The value of \((f \bullet g)(1)\) is 130.

Step-by-step solution

Substitute into f(x)

Given that \(f(2)=0\), substitute \(x=2\) into the function \(f(x)\) to find the value of \(b\). \(f(2) = 2(2)^3 - 14(2)^2 + b(2) - 3 = 0\) Simplify: \(16 - 56 + 2b - 3 = 0\) Combine like terms: \(-43 + 2b = 0\) Solve for \(b\): \(2b = 43\) \(b = 21.5\)

Utilize the given zero in g(x)

Since \(3 - i\) is a zero of \(g(x)\) and it's a polynomial with real coefficients, its conjugate, \(3 + i\), must also be a zero. This implies \(g(x)\) can be written as: \(g(x) = (x - (3 - i))(x - (3 + i))(x - k)\) for some real number \(k\).

Expand the conjugate pair

Multiply the factors associated with the conjugate pairs: \((x - (3 - i))(x - (3 + i)) = [(x - 3) + i][(x - 3) - i] = (x - 3)^2 - i^2 = x^2 - 6x + 9 + 1 = x^2 - 6x + 10\)

Form the complete function g(x)

Given \(g(x) = x^3 + c x^2 - 8 x + 30\), set it equal to the expanded product and solve for \(k\): \(g(x) = (x^2 - 6x + 10)(x - k)\) Expand: \(x^3 - kx^2 - 6x^2 + 6kx + 10x - 10k\) Compare coefficients: \(1. \, x^3\): No new information. \(2. \, -k - 6 = c\) so, \(c = -k - 6\), \(3. \, 6k + 10 = -8\) solve for \(k\): \(6k + 10 = -8 \ 6k = -18 \ k = -3\)

Solve for c

Using \(c = -k - 6\): \(c = -(-3) - 6 = 3 - 6 = -3\)

Evaluate \((f \bullet g) (1)\)

Substitute \(b\) and \(c\) back into the functions and evaluate: \(f(x) = 2 x^3 - 14 x^2 + 21.5 x - 3\) \(g(x) = x^3 - 3 x^2 - 8 x + 30\) Then \((f \bullet g) (1) = f(1) \bullet g(1)\) Evaluate \(f(1)\): \(f(1) = 2(1^3) - 14(1^2) + 21.5(1) - 3 = 2 - 14 + 21.5 - 3 = 6.5\) Evaluate \(g(1)\): \(g(1) = (1^3) - 3(1^2) - 8(1) + 30 = 1 - 3 - 8 + 30 = 20\) Now calculate the product: \((f \bullet g) (1) = 6.5 \bullet 20 = 130\)

Key concepts

zeros of polynomials

In any polynomial, the 'zeros' or 'roots' are the values of x that make the polynomial equal to zero. For the given functions, a zero of the polynomial is identified by solving the equation for x when it equals zero. In the exercise, we started with the polynomial function f(x) and were given that f(2) = 0, meaning that x = 2 is a zero of f(x).
Additionally, for a polynomial g(x) with a complex root (such as 3 - i), its conjugate (3 + i) must also be a zero if g(x) has real coefficients. This is because complex roots of polynomials with real coefficients always occur in conjugate pairs.

complex conjugates

Complex conjugates are pairs of complex numbers where the imaginary part has the opposite sign. For instance, if 3 - i is a root of a polynomial, its conjugate is 3 + i.
This principle is important when dealing with polynomials that have real coefficients because complex roots will not appear alone; they always come in these pairs. In our exercise, this property helped us determine additional roots of the polynomial g(x) simply through understanding that g(x) has real coefficients.

substitution method

The substitution method is used to solve equations by substituting a specific value of x into the polynomial to find missing coefficients. In the exercise, we used this method to find the value of b in f(x).
We substituted x = 2 (since f(2) = 0) into the equation f(x) and solved for b. This method helps to simplify complex polynomial expressions by substituting known values and making the equation easier to manage.

polynomial multiplication

Polynomial multiplication involves multiplying each term in one polynomial with each term in the other polynomial. In this exercise, we expanded the product of conjugate pair factors such as \( (x - (3 - i))(x - (3 + i)) \).
The steps to multiply these are:

• First, distribute each term: \( (x - 3 + i)(x - 3 - i) \)
• Then, use the FOIL method (First, Outside, Inside, Last) to simplify: \ x^2 - 6x + 10 \.

This process helps to simplify the structure of polynomial expansions and makes it more straightforward to solve for roots and coefficients.