Problem 77 - Beginning and Intermediate Algebra

Question

Find the indicated values for the following polynomial functions. \(f(x)=x^{2}+10 x+21 .\) Find \(x\) so that $f(x)=0$$

Step-by-Step Solution

Answer

The roots of the given polynomial function \(f(x) = x^2 + 10x + 21\) are \(x = -3\) and \(x = -7\).

Step 1: Identify the quadratic function coefficients

We have a quadratic function in the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are the coefficients. For our given function, \(f(x) = x^2 + 10x + 21\), the coefficients are: \(a = 1\), \(b = 10\), and \(c = 21\).

Step 2: Factor the quadratic polynomial

We will try factoring the quadratic function, \(f(x) = x^2 + 10x + 21\), to find the roots. If we can factor it, it will be in the form \((x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots. Looking for two integers whose product is 21 and whose sum is 10, we find that: \(3 \times 7 = 21\) and \(3 + 7 = 10\). So, we can write the quadratic function as: \(f(x) = (x + 3)(x + 7)\).

Step 3: Find the roots

Now we have factored the quadratic function into \((x + 3)(x + 7)\). To find the roots, we will set each factor equal to 0 and solve for x. First factor: \(x + 3 = 0 \Rightarrow x = -3\) Second factor: \(x + 7 = 0 \Rightarrow x = -7\)

Step 4: Write the final solution

We have found the roots of the polynomial function \(f(x) = x^2 + 10x + 21\) by factoring and solving for x. The roots are \(x = -3\) and \(x = -7\). So, the function will be 0 for these values of x.

Key Concepts

Factoring Polynomials

Factoring polynomials is a key technique used in solving quadratic equations. When you have a polynomial like the quadratic function given in the exercise, finding its factors can simplify the problem to a point where the roots become apparent. The specific process of factoring involves breaking down the polynomial into two binomial expressions.
In our example, the quadratic function represented by \( x^2 + 10x + 21 \) was rewritten as \((x + 3)(x + 7)\).

• These factors are achieved by identifying two numbers that multiply to the constant term \(c\) (21 in this case), and
• simultaneously add up to the linear coefficient \(b\) (10 here).

This method allows us to write the equation as a product of two simpler terms. The factors \((x + 3)\) and \((x + 7)\) are instrumental because they directly relate to the solutions or roots of the quadratic equation.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, and are typically modeled in the standard form \(f(x) = ax^2 + bx + c\).
They graph as a parabola and involve squared variables.Key features of quadratic functions include:

• Vertex: The highest or lowest point of the parabola, depending on whether the parabola opens upwards or downwards.
• Axis of Symmetry: A vertical line passing through the vertex, which divides the parabola into symmetrical halves.
• Roots or Zeros: The x-values where the function equals zero. These are the solutions to the equation.

For the provided quadratic \( x^2 + 10x + 21 \), it's crucial to understand that the equation represents a parabola with potential intersections at the x-axis.
This significance is essential for visualizing solutions—also known as the roots—of the equation, when \( f(x) = 0 \).

Roots of Equations

Roots of equations are the values of the variable that make the equation true. In the context of quadratic equations, they correspond to the x-values where the function intersects the x-axis. For the function \( f(x) = x^2 + 10x + 21 \), the roots are found by setting the equation equal to zero and solving for \( x \). As derived through factoring, we have the factors \((x + 3)(x + 7)\), leading to roots at \( x = -3 \) and \( x = -7 \).

• Roots can be real or complex numbers, and the method of finding them often dictates their nature.
• In this factorization case, both roots are real and distinct, showing distinct crossing points of the parabola at the x-axis.

Remember, identifying the roots gives meaningful insights into the graph of the function and helps determine the function's zeros, which are critically valued in various applications of mathematics.