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Chapter 11: Problem 53

Write a third-degree equation having the given numbers as solutions. $$ -2,1,5 $$

Short Answer

Expert verified
The third-degree polynomial is f(x) = x^3 - 4x^2 - 7x + 10.

Step by step solution

01

- Identify the Roots

Given the roots of the polynomial are -2, 1, and 5.
02

- Write the Factors

Since -2, 1, and 5 are roots of the polynomial, the corresponding factors of the polynomial are f(x) = (x + 2)(x - 1)(x - 5).
03

- Multiply the Factors

Multiply the factors to get the polynomial: First, multiply \( (x + 2)(x - 1)\):\[(x + 2)(x - 1) = x^2 - x + 2x - 2 = x^2 + x - 2\] Now multiply this result by \( (x - 5)\):\[ (x^2 + x - 2)(x - 5) = x^3 - 5x^2 + x^2 - 5x - 2x + 10 = x^3 - 4x^2 - 7x + 10 \]
04

- Write the Final Equation

The third-degree polynomial having the given numbers as solutions is f(x) = x^3 - 4x^2 - 7x + 10.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial roots
Roots of a polynomial are values of the variable that make the polynomial equal to zero. For example, in the polynomial equation \(f(x) = (x + 2)(x - 1)(x - 5)\), the roots are -2, 1, and 5. This means if you substitute any of these roots into \(f(x)\), the result will be zero. Think of roots as the points where the graph of the polynomial intersects the x-axis. These roots are helpful in constructing the polynomial.
Factoring polynomials
Factoring a polynomial means breaking it down into simpler polynomials that, when multiplied together, give back the original polynomial. Given the roots -2, 1, and 5, we can write the polynomial in factored form as \(f(x) = (x + 2)(x - 1)(x - 5)\). Notice that each factor corresponds to one of the roots. This process captures the idea that by using the roots, you can build or reconstruct the original equation.
Multiplying polynomials
Multiplying polynomials involves using distributive property to expand the factors step by step. Starting with \((x + 2)(x - 1)\), multiply each term inside the parentheses: \[ (x + 2)(x - 1) = x^2 - x + 2x - 2 = x^2 + x - 2 \] Next, multiply this result by \((x - 5)\): \( (x^2 + x - 2)(x - 5) = x^3 - 5x^2 + x^2 - 5x - 2x + 10 = x^3 - 4x^2 - 7x + 10 \) This gives you the polynomial in expanded form.
Writing polynomial equations
Writing polynomial equations involves determining the polynomial that has the given roots. In our case, we have the roots -2, 1, and 5. First, write the polynomial in its factored form using these roots: \(f(x) = (x + 2)(x - 1)(x - 5)\). Then, by multiplying the factors, which we did in earlier sections, we get the expanded form: \[ f(x) = x^3 - 4x^2 - 7x + 10\]. This final form represents the polynomial equation with the given roots.

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Most popular questions from this chapter

Solve by completing the square. Show your work. $$ x^{2}+16 x+15=0 $$

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ f(x)=x^{2}-8 x-10 $$

For each of the following, write the equation of the parabola that has the shape of \(f(x)=2 x^{2}\) or \(g(x)=-2 x^{2}\) and has a maximum value or \(a\) minimum value at the specified point. Minimum: \((-4,0)\)

Review solving systems of three equations in three unknowns. Solve. $$ \begin{aligned} &1.5=c\\\ &52.5=25 a+5 b+c\\\ &7.5=4 a+2 b+c \end{aligned} $$

Solve. $$ \left(y+\frac{3}{4}\right)^{2}=\frac{17}{16} $$
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