Math Problem Statement
Solution
Let's analyze the image and solve the polynomial problem step by step.
We are given the polynomial:
The task is to determine the correct remainder when this polynomial is divided by various polynomials, which is done using the Remainder Theorem.
Expanding :
First, expand the product:
Step 1: Expand :
Step 2: Multiply by :
Now, add to this result:
Thus, the polynomial is:
Now, let's check the remainder for each option:
- First option: Remainder of when divided by .
The polynomial . Since this matches the form of , the remainder should be zero, but the option suggests it is . This option is incorrect.
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Second option: Same division, but the remainder is given as . This is also incorrect since the remainder is zero.
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Third option: Remainder of divided by .
Since and divides into , the remainder is found by performing polynomial division. The remainder is , making this option correct.
- Fourth option: Remainder of divided by .
Using the Remainder Theorem, substitute into :
So, the remainder is 0, not , making this option incorrect.
Correct answer: The third option is correct.
Do you want any further details or clarifications? Here are some related questions:
- What is the Remainder Theorem, and how is it applied?
- How do you perform polynomial division step by step?
- Can polynomial factorization help in finding remainders faster?
- What is synthetic division, and when is it useful?
- What are some common mistakes to avoid when using the Remainder Theorem?
Tip: When dividing polynomials, always check for factorization shortcuts before performing full division.
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Math Problem Analysis
Mathematical Concepts
Polynomial Division
Remainder Theorem
Algebra
Formulas
Remainder Theorem: p(x) = (x + 2)(x^2 - 1)
Theorems
Polynomial Remainder Theorem
Suitable Grade Level
Grades 10-12