Math Problem Statement

Find all real zeros and the factored form of the polynomial P(x) = x^3 + 11x^2 + 8x - 20.

Solution

The given polynomial is P(x)=x3+11x2+8x−20P(x) = x^3 + 11x^2 + 8x - 20. We need to find all the real zeros of this polynomial and then write it in factored form.

Step 1: Try Rational Root Theorem

The Rational Root Theorem suggests that possible rational roots are factors of the constant term (-20) divided by factors of the leading coefficient (1). So the possible rational roots are: ±1,±2,±4,±5,±10,±20\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20

Step 2: Synthetic Division or Substitution

Let's check possible roots by substitution or synthetic division.

Test x=1x = 1:

P(1)=13+11(12)+8(1)−20=1+11+8−20=0P(1) = 1^3 + 11(1^2) + 8(1) - 20 = 1 + 11 + 8 - 20 = 0 So, x=1x = 1 is a root.

Step 3: Polynomial Division

Now, divide P(x)P(x) by (x−1)(x - 1) using synthetic or long division.

Performing synthetic division of P(x)=x3+11x2+8x−20P(x) = x^3 + 11x^2 + 8x - 20 by (x−1)(x - 1):

1 & 1 & 11 & 8 & -20 \\ & & 1 & 12 & 20 \\ \hline & 1 & 12 & 20 & 0 \\ \end{array}$$ The quotient is $$x^2 + 12x + 20$$. ### Step 4: Solve the Quadratic Now, solve $$x^2 + 12x + 20 = 0$$ using the quadratic formula: $$x = \frac{-12 \pm \sqrt{12^2 - 4(1)(20)}}{2(1)} = \frac{-12 \pm \sqrt{144 - 80}}{2} = \frac{-12 \pm \sqrt{64}}{2} = \frac{-12 \pm 8}{2}$$ Thus, $$x = \frac{-12 + 8}{2} = -2$$ or $$x = \frac{-12 - 8}{2} = -10$$. ### Step 5: Write the Factored Form The real zeros are $$x = 1, -2, -10$$. Therefore, the polynomial in factored form is: $$P(x) = (x - 1)(x + 2)(x + 10)$$ Would you like more details or have any questions? --- Here are 5 related questions: 1. What does the Rational Root Theorem state, and how is it applied? 2. Can synthetic division be used for non-rational roots? 3. How do we perform long division of polynomials? 4. What is the quadratic formula and when do we use it? 5. What are the differences between real and complex zeros of a polynomial? **Tip**: When checking for rational roots, always try simple numbers like $$\pm 1$$ first, as they are easier to compute!

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Math Problem Analysis

Mathematical Concepts

Polynomial Zeros
Factoring Polynomials
Quadratic Equations

Formulas

Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a

Theorems

Rational Root Theorem

Suitable Grade Level

Grades 10-12

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