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Chapter 11: Problem 55

Write a third-degree equation having the given numbers as solutions. $$ -1,0,3 $$

Short Answer

Expert verified
The third-degree equation is \[ x^3 - 2x^2 - 3x = 0. \]

Step by step solution

01

Understand the solutions

The given solutions of the third-degree equation are -1, 0, and 3.
02

Set up the equation using the solutions

Since -1, 0, and 3 are the solutions, the factors of the polynomial are \(x + 1\), \(x\), and \(x - 3\).
03

Form the polynomial by multiplying the factors

The polynomial can be written by multiplying the factors together: \[(x + 1) (x) (x - 3)\]
04

Multiply the first two factors

Multiply \(x + 1\) and \(x\): \[x(x + 1) = x^2 + x\]
05

Multiply the resulting expression by the next factor

Next, multiply \(x^2 + x\) by \(x - 3\): \[ (x^2 + x)(x - 3) = x^3 - 3x^2 + x^2 - 3x = x^3 - 2x^2 - 3x \]

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Key Concepts

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

polynomial equations
Polynomial equations are equations involving polynomials, which are expressions made up of variables and coefficients.
These variables only involve whole number exponents, and no variables must be in the denominator of any term.
A third-degree polynomial is a polynomial of degree three, which means that its highest power of the variable is three.
For example, an equation like x^3 - 2x^2 - 3x = 0 is a third-degree polynomial equation.
The solutions to the polynomial are the values of the variable that make the polynomial equal to zero.
In the example above, the solutions are -1, 0, and 3, since substituting any of these values for x will satisfy the equation.
factors of a polynomial
Factors of a polynomial are algebraic expressions that can be multiplied together to get the original polynomial.
If we know the solutions to a polynomial equation, we can easily write the polynomial in its factored form.
For example, with solutions (-1, 0, 3), the factors are (x + 1), (x), and (x - 3). Multiplying these factors together, we get the polynomial: (x + 1) (x) (x - 3) = 0. Writing the polynomial in its factored form helps us see its roots more easily.
multiplying polynomials
When multiplying polynomials, we use the distributive property to make sure each term in one polynomial is multiplied by each term in the other polynomial.
Let鈥檚 multiply (x + 1) and (x) first: \(x(x + 1) = x^2 + x\).
Next, we'll multiply the result by the final factor (x - 3). We distribute x^2 + x across each term in (x - 3): (x^2 + x)(x - 3) = x^3 - 3x^2 + x^2 - 3x. Combining like terms, we get x^3 - 2x^2 - 3x.
The product of these factors gives us the original polynomial, completing the process.

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