Math Problem Statement

Factor​ f(x) into linear factors given that k is a zero of​ f(x). f(x)=4x3− 9x2−58x+15​; k=5

Solution

To factor f(x)=4x39x258x+15f(x) = 4x^3 - 9x^2 - 58x + 15 into linear factors given k=5k = 5 is a zero, follow these steps:


Step 1: Synthetic Division

Since k=5k = 5 is a zero, divide f(x)f(x) by (x5)(x - 5) using synthetic division.

Synthetic Division Setup:

5 & 4 & -9 & -58 & 15 \\ & & 20 & 55 & -15 \\ \hline & 4 & 11 & -3 & 0 \\ \end{array}$$ #### Result: The quotient is $$4x^2 + 11x - 3$$ with a remainder of 0, confirming that $$x - 5$$ is a factor. Thus: $$f(x) = (x - 5)(4x^2 + 11x - 3)$$ --- ### **Step 2: Factor the Quadratic $$4x^2 + 11x - 3$$** We now factor $$4x^2 + 11x - 3$$ using the splitting-the-middle method. 1. Multiply the leading coefficient and the constant: $$4 \times (-3) = -12$$. 2. Find two numbers that multiply to $$-12$$ and add to $$11$$: $$12$$ and $$-1$$. 3. Rewrite the middle term $$11x$$ as $$12x - x$$: $$4x^2 + 12x - x - 3$$ 4. Group terms and factor: $$(4x^2 + 12x) - (x + 3) = 4x(x + 3) - 1(x + 3)$$ $$= (4x - 1)(x + 3)$$ --- ### **Step 3: Write the Complete Factorization** Now we have: $$f(x) = (x - 5)(4x - 1)(x + 3)$$ --- ### Final Answer: $$f(x) = (x - 5)(4x - 1)(x + 3)$$ --- Would you like further details on any step? Here are five related questions to deepen your understanding: 1. How does synthetic division confirm a zero of a polynomial? 2. What alternative methods can you use to factor quadratics like $$4x^2 + 11x - 3$$? 3. How do you verify that the factorization is correct? 4. What is the significance of all factors being linear in this context? 5. Can $$f(x)$$ have other real or complex roots? **Tip**: Always check your factorization by multiplying the factors back together to ensure they reconstruct the original polynomial.

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Math Problem Analysis

Mathematical Concepts

Polynomial Division
Factoring Polynomials
Synthetic Division
Quadratic Equations

Formulas

Synthetic Division: Dividend ÷ (x - c)
Factoring Quadratics: ax^2 + bx + c = a(x - p)(x - q)

Theorems

Remainder Theorem
Factor Theorem

Suitable Grade Level

Grades 10-12