Textbook Question
Use synthetic division to find ƒ(2). ƒ(x)=2x3-3x2+7x-12
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4. Polynomial Functions
Dividing Polynomials
2:41 minutesProblem 45
Textbook Question
For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = 6x4 + x3 - 8x2 + 5x+6; k=1/2
Verified step by step guidance1
Recall the Remainder Theorem: For a polynomial ƒ(x), the remainder when divided by (x - k) is equal to ƒ(k). So, to find ƒ(k), we simply evaluate the polynomial at x = k.
Write down the polynomial function and substitute x with k = \(\frac{1}{2}\):
\[ƒ\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right)^4 + \left(\frac{1}{2}\right)^3 - 8\left(\frac{1}{2}\right)^2 + 5\left(\frac{1}{2}\right) + 6\]
Calculate each term separately:
- Compute \(6\left(\frac{1}{2}\right)^4\)
- Compute \(\left(\frac{1}{2}\right)^3\)
- Compute \(-8\left(\frac{1}{2}\right)^2\)
- Compute \(5\left(\frac{1}{2}\right)\)
- The constant term is 6
Add all the computed values together to find the value of ƒ\(\left\)(\(\frac{1}{2}\)\(\right\)). This sum represents the remainder when ƒ(x) is divided by \(x - \frac{1}{2}\).
Thus, the value of ƒ(k) is the remainder according to the Remainder Theorem, completing the problem.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Functions
A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. Understanding the structure of polynomials helps in evaluating them at specific values and applying theorems related to their behavior.
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The Remainder Theorem states that when a polynomial ƒ(x) is divided by (x - k), the remainder is equal to ƒ(k). This allows us to find the value of the polynomial at k by simply evaluating ƒ(k), without performing full polynomial division.
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Evaluating a polynomial at a specific value involves substituting the value for the variable and simplifying the expression. This process is essential for applying the Remainder Theorem and finding the remainder or function value efficiently.
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