If y = 1 + √3 + √4, then the value of 2y4 - 8y3 - 6y2 + 28y - 84 is:
40√3
Evaluating the Polynomial Expression with Radicals
We are given the value of \(y\) as \(y = 1 + \sqrt{3} + \sqrt{4}\) and asked to find the value of the polynomial expression \(2y^4 - 8y^3 - 6y^2 + 28y - 84\).
Step 1: Simplify the Expression for \(y\)
The first step is to simplify the given value of \(y\):
\(y = 1 + \sqrt{3} + \sqrt{4}\)
Since \(\sqrt{4} = 2\), we can substitute this into the expression for \(y\):
\(y = 1 + \sqrt{3} + 2\)
\(y = 3 + \sqrt{3}\)
Step 2: Find a Polynomial Equation Satisfied by \(y\)
To make the evaluation of the higher-degree polynomial easier, we can find a simpler polynomial equation that \(y\) satisfies. From \(y = 3 + \sqrt{3}\), we can isolate the radical term:
\(y - 3 = \sqrt{3}\)
Now, square both sides of the equation to eliminate the square root:
\((y - 3)^2 = (\sqrt{3})^2\)
Expanding the left side gives:
\(y^2 - 2(y)(3) + 3^2 = 3\)
\(y^2 - 6y + 9 = 3\)
Rearrange the terms to form a quadratic equation:
\(y^2 - 6y + 9 - 3 = 0\)
\(y^2 - 6y + 6 = 0\)
This equation is satisfied by the given value of \(y\).
Step 3: Simplify the Given Polynomial Expression Using the Found Equation
We want to evaluate the polynomial \(P(y) = 2y^4 - 8y^3 - 6y^2 + 28y - 84\). Since \(y\) satisfies \(y^2 - 6y + 6 = 0\), we know that \(y^2 = 6y - 6\). We can use this relationship to reduce the degree of the polynomial.
Method 1: Repeated Substitution
We can express higher powers of \(y\) in terms of \(y\) and a constant:
- \(y^2 = 6y - 6\)
- \(y^3 = y \cdot y^2 = y(6y - 6) = 6y^2 - 6y\). Substitute \(y^2 = 6y - 6\) again: \(y^3 = 6(6y - 6) - 6y = 36y - 36 - 6y = 30y - 36\).
- \(y^4 = y \cdot y^3 = y(30y - 36) = 30y^2 - 36y\). Substitute \(y^2 = 6y - 6\) again: \(y^4 = 30(6y - 6) - 36y = 180y - 180 - 36y = 144y - 180\).
Now, substitute these expressions back into the original polynomial \(P(y)\):
\(P(y) = 2(144y - 180) - 8(30y - 36) - 6(6y - 6) + 28y - 84\)
Expand and group terms:
\(P(y) = 288y - 360 - 240y + 288 - 36y + 36 + 28y - 84\)
Group terms with \(y\): \(288y - 240y - 36y + 28y = (288 - 240 - 36 + 28)y = (48 - 36 + 28)y = (12 + 28)y = 40y\)
Group constant terms: \(-360 + 288 + 36 - 84 = -72 + 36 - 84 = -36 - 84 = -120\)
So, the simplified polynomial is \(P(y) = 40y - 120\).
Method 2: Polynomial Long Division
Alternatively, we can perform polynomial long division of \(P(y) = 2y^4 - 8y^3 - 6y^2 + 28y - 84\) by \(Q(y) = y^2 - 6y + 6\). Since \(Q(y) = 0\) for the value of \(y\), the value of \(P(y)\) will be equal to the remainder of this division.
| Step | Operation | Intermediate Result |
|---|---|---|
| 1 | Divide \(2y^4\) by \(y^2\): \(2y^2\) Multiply \(2y^2(y^2 - 6y + 6)\): \(2y^4 - 12y^3 + 12y^2\) Subtract from original polynomial |
\(4y^3 - 18y^2 + 28y - 84\) |
| 2 | Divide \(4y^3\) by \(y^2\): \(4y\) Multiply \(4y(y^2 - 6y + 6)\): \(4y^3 - 24y^2 + 24y\) Subtract from remainder |
\(6y^2 + 4y - 84\) |
| 3 | Divide \(6y^2\) by \(y^2\): \(6\) Multiply \(6(y^2 - 6y + 6)\): \(6y^2 - 36y + 36\) Subtract from remainder |
\(40y - 120\) |
The remainder of the division is \(40y - 120\). Thus, \(P(y) = (y^2 - 6y + 6)(2y^2 + 4y + 6) + 40y - 120\). Since \(y^2 - 6y + 6 = 0\), we have \(P(y) = 0 \cdot (2y^2 + 4y + 6) + 40y - 120 = 40y - 120\).
Step 4: Substitute the Value of \(y\) into the Simplified Expression
Now substitute the value \(y = 3 + \sqrt{3}\) into the simplified expression \(40y - 120\):
Value \( = 40(3 + \sqrt{3}) - 120\)
Distribute the 40:
Value \( = 40 \times 3 + 40 \times \sqrt{3} - 120\)
Value \( = 120 + 40\sqrt{3} - 120\)
The constant terms cancel out:
Value \( = 40\sqrt{3}\)
Conclusion on Polynomial Value
The value of the expression \(2y^4 - 8y^3 - 6y^2 + 28y - 84\) when \(y = 1 + \sqrt{3} + \sqrt{4}\) is \(40\sqrt{3}\).
Revision Table: Key Algebra Concepts
| Concept | Description | Relevance to Problem |
|---|---|---|
| Simplifying Radicals | Finding the simplest form of a square root, like \(\sqrt{4}=2\). | Initial step to simplify \(y\). |
| Forming a Polynomial Equation | Manipulating an expression with radicals to find an equation (e.g., quadratic) the variable satisfies. | Crucial step \(y-3=\sqrt{3} \implies y^2-6y+6=0\). |
| Polynomial Remainder Theorem principle | If \(P(y) = Q(y)S(y) + R(y)\) and \(Q(y) = 0\) for a value of \(y\), then \(P(y) = R(y)\). | Justification for using polynomial division remainder or substitution method. |
| Polynomial Simplification | Reducing the degree of a polynomial expression using a known relationship between powers of the variable. | Main technique used (substitution or division). |
| Substitution Method | Replacing powers of the variable (\(y^2, y^3, y^4\)) with equivalent simpler expressions. | Method used to reduce the polynomial degree step-by-step. |
| Polynomial Long Division | An algorithm to divide one polynomial by another. | Alternative method to find the remainder, which is the simplified expression value. |
Additional Information: Handling Expressions with Multiple Radicals
For expressions like \(y = a + \sqrt{b} + \sqrt{c}\), simplifying and finding a polynomial equation usually involves isolating one radical, squaring, then isolating the remaining radical, and squaring again. For \(y = 1 + \sqrt{3} + \sqrt{4}\), it simplified to \(y = 3 + \sqrt{3}\), a form with only one radical, which is simpler.
If \(y = a + \sqrt{b} + \sqrt{c}\), the process typically looks like this:
- Start with \(y - a = \sqrt{b} + \sqrt{c}\).
- Square both sides: \((y - a)^2 = (\sqrt{b} + \sqrt{c})^2 = b + c + 2\sqrt{bc}\).
- Rearrange to isolate the remaining radical: \((y - a)^2 - b - c = 2\sqrt{bc}\).
- Square both sides again: \( ((y - a)^2 - b - c)^2 = (2\sqrt{bc})^2 = 4bc \).
Expanding this last equation results in a polynomial in \(y\) with integer coefficients. This polynomial will be of degree 4. Once you have such a polynomial \(Q(y) = 0\), you can use polynomial division or substitution to simplify the given expression \(P(y)\) to its remainder when divided by \(Q(y)\).
The key takeaway is that finding a polynomial equation satisfied by the variable simplifies evaluating complex polynomial expressions involving that variable, especially when the variable contains radical terms.
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