7B - Identifying factors of a polynomial
The factor/remainder theorems:
■ The factor theorem state that if \(P(a)=0\), then \((x-a)\) is a factor of \(P(x)\).
7B - Example 1: Using the factor theorem to find linear factors of a polynomial
Using the factor theorem, identify the linear factors of the following polynomial: \[P(x)=x^3+4x^2+x-6\]
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7B - Example 1: Video solution
7B - Example 1: Practice
Question 1: ABC Question 2: ABC 7B - Example 1: Solutions
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7B - Example 2: Applying the factor theorem to find coefficients of a polynomial
Find the values of \(a\) and \(b\) given that \((x-3)\) and \((x+2)\) are factors of the polynomial \[P(x)=x^3+6x^2+ax+b\]
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7B - Example 2: Video solution
7B - Example 2: Practice
Question 1: ABC Question 2: ABC 7B - Example 2: Solutions
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■ The remainder theorem state that when \(P(x)\) is divided by \((ax+b)\), the remainder is given by \(P(-\frac{b}{a})=r\)
- If \(r=0\) then \((ax+b)\) is a factor of \(P(x)\) as described by the factor theorem discussed above.
7B - Example 3: Applying the remainder theorem to find coefficients of a polynomial
When \(P(x)=x^3+3x^2+7x-b\) is divided by \((x-2)\), the remainder is 20. Find the value of \(b\).
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7B - Example 3: Video solution
7B - Example 3: Practice
Question 1: ABC Question 2: ABC 7B - Example 3: Solutions
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7B - Example 4: Applying the remainder theorem to find coefficients of a polynomial
When \(P(x)=x^3+ax^2+x+b\) is divided by \((x+1)\) or \((x-2)\), the remainder is 0. Find the values of \(a\) and \(b\).
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7B - Example 4: Video solution
7B - Example 4: Practice
Question 1: ABC Question 2: ABC 7B - Example 4: Solutions
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7B - Example 5: Applying the remainder theorem to find coefficients of a polynomial
When \(P(x)=x^3+ax^2+bx+3\), is divided by \((x+1)\) the remainder is 1 and when \(P(x)\) is divided by \((x-1)\) the remainder is 11. Find the values of \(a\) and \(b\).
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7B - Example 5: Video solution
7B - Example 5: Practice
Question 1: ABC Question 2: ABC 7B - Example 5: Solutions
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The rational root theorem:
7B - Content video: The rational root theorem
This video covers the theory and an example relating to the rational root theorem.
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