Question

Find a polynomial function with the given real zeros whose graph contains the given point. Zeros: -1 (multiplicity 2), 1 (multiplicity 2) Degree 4 Point: (-2,45)

Short answer

The polynomial function is f(x) = 5(x^4 - 2x^2 + 1).

Step-by-step solution

Identify the Polynomial Form

Given the zeros -1 (multiplicity 2) and 1 (multiplicity 2), the polynomial can be written as: f(x) = a(x + 1)^2(x - 1)^2

Expand the Polynomial

Expand (x + 1)^2(x - 1)^2: (x + 1)^2 = x^2 + 2x + 1 (x - 1)^2 = x^2 - 2x + 1 Using distribution: f(x) = a(x^2 + 2x + 1)(x^2 - 2x + 1)

Simplify the Polynomial

Use the distribution to expand and simplify the polynomial: f(x) = a(x^2(x^2 - 2x + 1) + 2x(x^2 - 2x + 1) + 1(x^2 - 2x + 1)) f(x) = a(x^4 - 2x^3 + x^2 + 2x^3 - 4x^2 + 2x + x^2 - 2x + 1) f(x) = a(x^4 - 2x^2 + 1)

Determine the Coefficient 'a'

Substitute the point (-2, 45) into the polynomial to find 'a': 45 = a((-2)^4 - 2(-2)^2 + 1) 45 = a(16 - 8 + 1) 45 = a(9) a = 5

Write the Final Polynomial

Substitute 'a' back into the polynomial: f(x) = 5(x^2 + 2x + 1)(x^2 - 2x + 1) After expansion and simplification: f(x) = 5(x^4 - 2x^2 + 1)

Key concepts

real zeros

In polynomial functions, real zeros are the x-values where the polynomial equals zero. These are the points where the graph of the function crosses or touches the x-axis. Identifying the real zeros of a polynomial is essential for sketching its graph and solving equations.

**Key Points to Remember about Real Zeros**

• Real zeros are found by solving the equation f(x) = 0.
• Each zero represents an x-value where the polynomial crosses the x-axis.
• Real zeros can be repeated, impacting the shape of the graph.

In the given exercise, the polynomial function has real zeros at -1 and 1. These zeros indicate that f(x) touches the x-axis at these points.

multiplicity of zeros

Multiplicity of zeros refers to how many times a particular zero appears in a polynomial. It indicates whether the graph of the function merely touches the x-axis at that point or actually crosses it.

**Understanding Multiplicity**

• A zero with an **odd multiplicity** causes the graph to cross the x-axis.
• A zero with an **even multiplicity** causes the graph to touch but not cross the x-axis.
• Higher multiplicity flattens the graph at the zero.

For our exercise, the zeros -1 and 1 both have a multiplicity of 2. This means the graph touches the x-axis at these points but does not cross it.

polynomial expansion

Polynomial expansion involves distributing the factors of a polynomial to write it as a sum of its terms. This process helps in simplifying and solving polynomial equations.

**Steps in Polynomial Expansion**

• Expand any squared or higher power terms first.
• Use the distributive property to multiply the expanded terms.
• Combine like terms to simplify the expression.

In this exercise, we expand \((x + 1)^2(x - 1)^2\) starting with:

\((x + 1)^2 = x^2 + 2x + 1\)
\((x - 1)^2 = x^2 - 2x + 1\)

Using distribution, we get:
\(a(x^2 + 2x + 1)(x^2 - 2x + 1)\)

After expansion and combining like terms, the final polynomial is simplified.

finding coefficients

Finding the coefficients in a polynomial is crucial for writing the exact function. Coefficients are the constants that multiply the variables in each term of the polynomial.

**Process of Finding Coefficients**

• Form the expanded polynomial expression.
• Substitute the given point to solve for the leading coefficient.
• Multiply the polynomial by this coefficient.

In our exercise, we substitute (-2, 45) into the polynomial:

\(45 = a((-2)^4 - 2(-2)^2 + 1) = 9a \).
Solving for 'a' gives us: \ a = 5 \.

Therefore, the final polynomial with the correct coefficient is \ f(x) = 5(x^4 - 2x^2 + 1) \.