Problem 31
Question
Find the zeros of each function. State the multiplicity of multiple zeros. $$ y=2 x^{3}+x^{2}-x $$
Step-by-Step Solution
Verified Answer
The zeros of the function are \( x = 0 \) with multiplicity 1, \( x = \frac{1}{2} \) with multiplicity 1, and \( x = -1 \) with multiplicity 1.
1Step 1: Factor out the Greatest Common Factor (GCF)
Identify the greatest common factor of the terms in the polynomial. In this case, the GCF is x, since it is the factor that all terms have in common. Factor x out of the polynomial: \( y = x(2x^2 + x - 1) \).
2Step 2: Factor the Quadratic
Next, factor the quadratic expression inside the parentheses \(2x^2 + x - 1\). This does not factor neatly, so you may use the quadratic formula or attempt to find factors that work. In this case, the quadratic cannot be factored using integers, so it is irreducible over the integers.
3Step 3: Find the Zeros
Set each factor equal to zero and solve for x: For the linear factor, set \( x = 0 \), which gives the first zero.For the irreducible quadratic, since it cannot be factored, you would use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \(a = 2\), \(b = 1\), and \(c = -1\) to find the other two zeros.
4Step 4: Apply the Quadratic Formula
Apply the quadratic formula to find the remaining zeros: \( x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-1)}}{2(2)} = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm 3}{4} \). This gives us two additional zeros: \( x = \frac{1}{2} \) and \( x = -1 \).
5Step 5: State the Multiplicities
The zero \( x = 0 \) comes from the linear factor x, which has a multiplicity of 1. The zeros \( x = \frac{1}{2} \) and \( x = -1 \) are the solutions to the irreducible quadratic, which means they also have a multiplicity of 1 each, since they don't repeat in the factorization.
Key Concepts
Factoring PolynomialsQuadratic FormulaPolynomial Roots Multiplicity
Factoring Polynomials
Understanding how to factor polynomials is crucial for finding their zeros. In essence, factoring breaks down a complex expression into simpler parts (factors) that, when multiplied together, give back the original polynomial. It's similar to dismantling a Lego structure into the individual blocks it's made of.
When we factor, we look for common elements in each term of the polynomial, like the Greatest Common Factor (GCF). In the example, the GCF is the variable 'x', which each term has. By factoring 'x' out, we simplify the polynomial to its constituent parts, in this case, a linear factor and a quadratic expression.
If the quadratic expression is factorable, it can be broken down further into two binomials. However, not all quadratics are that easy to deal with, especially when they cannot be factored using integers, known as 'irreducible over the integers.' In such cases, other methods, such as the quadratic formula, come into play. Factoring is a powerful tool that enables us to rewrite complex polynomials in a form that reveals their zeros directly or makes it easier to apply other methods for finding the zeros.
When we factor, we look for common elements in each term of the polynomial, like the Greatest Common Factor (GCF). In the example, the GCF is the variable 'x', which each term has. By factoring 'x' out, we simplify the polynomial to its constituent parts, in this case, a linear factor and a quadratic expression.
If the quadratic expression is factorable, it can be broken down further into two binomials. However, not all quadratics are that easy to deal with, especially when they cannot be factored using integers, known as 'irreducible over the integers.' In such cases, other methods, such as the quadratic formula, come into play. Factoring is a powerful tool that enables us to rewrite complex polynomials in a form that reveals their zeros directly or makes it easier to apply other methods for finding the zeros.
Quadratic Formula
The quadratic formula provides a straightforward method for finding the zeros of quadratic expressions that do not factor neatly. The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where 'a', 'b', and 'c' are the coefficients of the quadratic expression in the standard form \( ax^2 + bx + c \).
It's like a magic key that unlocks the solutions to any quadratic equation, even when factoring seems impossible. By substituting the coefficients into the formula, we can find the two values of 'x' that make the quadratic expression zero.
In our exercise, we applied the quadratic formula to a quadratic that couldn't be factored using integers. By calculating under the square root \( b^2 - 4ac \), known as the discriminant, we could determine the nature of the roots and their exact values. When the discriminant is positive, we get two distinct real roots, just as we found in the exercise, leading us to the zeros \( x = \frac{1}{2} \) and \( x = -1 \).
It's like a magic key that unlocks the solutions to any quadratic equation, even when factoring seems impossible. By substituting the coefficients into the formula, we can find the two values of 'x' that make the quadratic expression zero.
In our exercise, we applied the quadratic formula to a quadratic that couldn't be factored using integers. By calculating under the square root \( b^2 - 4ac \), known as the discriminant, we could determine the nature of the roots and their exact values. When the discriminant is positive, we get two distinct real roots, just as we found in the exercise, leading us to the zeros \( x = \frac{1}{2} \) and \( x = -1 \).
Polynomial Roots Multiplicity
The concept of multiplicity refers to the number of times a particular root appears as a solution to a polynomial equation. A root's multiplicity affects the shape of the graph of the polynomial and provides insight into its behavior near that root.
For instance, if a root is repeated as a factor of the polynomial, we say it has a higher multiplicity. Simple roots, or roots of multiplicity one, are points where the graph crosses the x-axis only once. In our exercise, each zero found, namely \( x = 0 \), \( x = \frac{1}{2} \), and \( x = -1 \), appeared only once in the factorized form, which means each has a multiplicity of one.
Knowing the multiplicity is vital as it can tell us about the touchpoints and crossing points of the polynomial's graph. It also affects the derivative of the polynomial; for instance, if a polynomial has a double root, the derivative will have a zero at that same x-value as well.
For instance, if a root is repeated as a factor of the polynomial, we say it has a higher multiplicity. Simple roots, or roots of multiplicity one, are points where the graph crosses the x-axis only once. In our exercise, each zero found, namely \( x = 0 \), \( x = \frac{1}{2} \), and \( x = -1 \), appeared only once in the factorized form, which means each has a multiplicity of one.
Knowing the multiplicity is vital as it can tell us about the touchpoints and crossing points of the polynomial's graph. It also affects the derivative of the polynomial; for instance, if a polynomial has a double root, the derivative will have a zero at that same x-value as well.
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